Everton's newest recruit Gerard Deulofeu insists it was his love with the supporters that has brought him back.
Bolton have completed the signing of former Sheffield Wednesday striker Gary Madine and finalised their deal with 'keeper Ben Amos.
Koeman, 50, last month declared his intention to leave Rotterdam after three years in the dubout at De Kuip, and Feyenoord have now turned to former PSV Eindhoven and Vitesse Arnhem boss Rutten.
The 51-year-old, who will soon sign a one-year contract before taking over from July 1, told feyenoord.com: "The club has obviously done fantastically under Ronald Koeman in recent years.
"After some lean years they have built something up in Rotterdam, from both a financial and sporting perspective.
"The broadly shared ambition to continue on the successful path can be seen and felt in everything. I would like to contribute to that and I know I can."
Feyenoord are currently fourth in the Eredivisie table, 12 points behind leaders Ajax.
However, a second-placed finish offers a route into the Champions League through the summer qualifiers, and front-runners FC Twente are currently only four points better off than Feyenoord.
"The next few weeks are too important for Feyenoord to keep talking about me," Rutten added.
"I don't want to get in the way of Ronald Koeman and the squad or affect their concentration in any way.
"For that reason, I will stay in the shadows for the time being."
Technical director Martin van Geel said: "As people know, we preferred to have an experienced top Dutch coach for the first team after Koeman.
"Given all the demands we make on a coach the list of names you can draw up is certainly not too long. So it's great when you can reach agreement quickly with someone who is very high up the list.
"Fred Rutten is an experienced head coach who has demonstrated that he can improve players and often gets teams playing more attractive football.
"With that in mind, the club management agreed unanimously that he was the right man to take over from Koeman after a couple of good years."