Celtic manager Ronny Deila insists Virgil van Dijk does not want to leave the Scottish Premiership leaders in January.
Celtic boss Ronny Deila says keeping his best players will take priority over acquisitions in the transfer window.
The split comes after Saturday's 2-1 league defeat at Heerenveen left Twente, who won the title under McClaren in 2010, in fifth place in the Eredivisie.
McClaren held talks with president Joop Munsterman over the weekend, but the manager failed to receive the assurances for which he had hoped.
The 51-year-old, who was in his second spell in charge at the Stadium De Grolsch Veste, said in a statement: "After the defeat to Heerenveen on Saturday, I met the President on Sunday.
"We tried to find a solution to the difficult period the team has endured over the last four weeks.
"We discussed the expectations of the club, how European qualification was key each season and finishing in the top four.
"Despite being joint top at the winter break and even second four weeks ago, recent results and performances have seen major criticism, protest and demonstration from the supporters and the media.
"We talked about how we could change the situation and relieve the pressure on the team which has clearly affected its performance.
"Unfortunately, Saturday's result has seen further adverse reactions and with two home games coming up, we feel the team needs the support and backing from the supporters.
"I also needed assurances about my position now and in the future, which I appreciate in the present situation is difficult.
"After much discussion and deliberation, those assurances weren't forthcoming and we have agreed it's in the best interest of the club going forward that I leave.
"The club is bigger than any one individual and Twente is too big in my heart to stand in the way of its progress.
"I leave with the team with 10 games to go, still able to achieve its expectations of a top-four finish, which I am positive they will do.
"I leave a proud man to have managed Twente. It has brought me many good memories and one of the greatest days in my career when winning the title in 2010.
"The support received from the club, staff, players and fans has always been much appreciated and I leave sad and disappointed it has ended in these circumstances, but wish the staff and the team success in the future."
McClaren leaves with Twente only a point adrift of Vitesse Arnhem in fourth place and a further five behind leaders PSV Eindhoven.
However, they have not tasted victory since a 3-0 win at AZ Alkmaar on December 21, a run of six games either side of the mid-season break which has seen them collect just four points from a possible 18.
Twente chairman Joop Munsterman revealed the decision to leave the club rested with McClaren and he was disappointed to see the Englishman leave.
Munsterman said in a statement on the club's official website: "We believe that the disappointing performance of FC Twente under Steve McClaren could be reversed.
"Steve has however indicated that it is in the interest of everyone to leave the club immediately.
"We understand his decision and respect it. It is disappointing to say goodbye to the trainer under whose leadership FC Twente won the league title in 2010."
Munsterman also confirmed the team will be led by trainer Alfred Schreuder and assisted by the technical team of Boudewijn Pahlplatz and Youri Mulder.
Steve McClaren factfile:
1961: Born Fulford, York, May 3.
1978: Joins Hull City.
1985: After 178 games and 16 goals for the Tigers, joins Derby.
1988: After 25 games for the Rams - and eight on loan at Lincoln - moves to Bristol City.
1990: Joins Oxford.
1995: Becomes assistant to Jim Smith at Derby after quitting as Oxford youth team coach.
1999: Joins Manchester United as Sir Alex Ferguson's assistant manager.
2001: February - Member of new England manager Sven-Goran Eriksson's coaching staff.
March - Ferguson blocks any continued involvement for McClaren in the England set-up.
June - Becomes Middlesbrough manager.
2002: June - Part of England's coaching staff as England are knocked out by Brazil in the quarter-finals of the World Cup.
November - Leaves position as Eriksson's assistant to concentrate on Middlesbrough.
2004: February - Wins Carling Cup after 2-1 defeat of Bolton - the first major trophy in Boro's history.
May - Rejoins England coaching staff for Euro 2004.
2006: May 4: Named as Eriksson's successor as England manager.
May 10 - Middlesbrough beaten 4-0 by Sevilla in the UEFA Cup final.
August - Starts England tenure with 4-0 friendly win over European champions Greece.
2007: March - Pressure mounts as England draw with Israel and struggle against Andorra in Euro 2008 qualifiers.
October - England beat Estonia 3-0 at Wembley, but lose 2-1 away at Russia to leave qualification out of their hands.
November 21 - England lose 3-2 to Croatia at Wembley and fail to qualify for Euro 2008.
November 22 - Sacked by the Football Association after an emergency board meeting at Soho Square.
2008: June 20 - Returns to club management with Dutch club FC Twente.
2009: Leads Twente to second-placed finish in the Eredivisie and enjoys runs to the Dutch cup final and UEFA Cup knockout stage.
October - Signs contract extension after an excellent start to the 2009/10 season.
2010: May 2 - Leads Twente to their first ever Eredivisie title with a 2-0 win at NAC Breda on the final day of the season.
May 11 - Leaves Twente to join Bundesliga side Wolfsburg on a two-year deal.
2011: February 7 - Parts company with Wolfsburg following a string of poor results.
June 13 - Appointed manager of Nottingham Forest on a three-year contract, succeeding Billy Davies.
October 2 - Resigns following 3-1 defeat at home to Birmingham.
2012: January 5 - Appointed FC Twente boss for a second time.
May 6 - Twente finish sixth in the Eredivisie but lose to RKC Waalwijk in the Europa League play-offs.
2013: February 26 - Leaves Twente by mutual consent after a run of six games without a win sees them drop to fifth in the Eredivisie table.