Bayern Munich forward Toni Kroos left the door open for a switch to the Barclays Premier League in the summer.
Tim Sherwood claims the "silence is deafening" from the Tottenham board amid speculation he could be replaced as manager.
Keller, whose only previous Bundesliga coaching experience came in an ill-fated two-month spell in charge of Stuttgart early in the 2010/11 season, was at work today preparing the side for their DFB-Pokal tie with Mainz on Tuesday night after being called in to replace Stevens.
The Dutch coach was fired after a run of six games without a win left the Royal Blues seventh in the table going into the second half of the season.
Keller, who moves up from the position of under-17 coach, inherits a side who, despite their recent league woes, have reached the last 16 of the Champions League in style, beating Arsenal to top spot in their group, and who are nonetheless only five points adrift of the top three.
"We are definitely planning with Jens Keller until the end of the season," club chairman Clemens Tonnies told Sport1 television.
Tonnies explained how the decision to dismiss Stevens, voted by the club's fans as their coach of the century recently, had to be taken due to a recent dip in form.
"We had to react," he said. "Huub Stevens is a great guy and he sacrificed his personal interests for the good of the team.
"He has admitted that he was no longer getting through to the team.
"I don't know exactly what has happened, but after the Leverkusen game, some kind of change took place."
The final straw was yesterday's 3-1 defeat to Freiburg which saw the club drop to their lowest position of the season, and out of the qualifying berths for European football next season.
Had Schalke won that game, Stevens would still be in his job now, revealed Tonnies.
"Of course you don't like to sack a coach just before Christmas," he said. "Had we won the Freiburg game, then we would probably have waited."
Stevens led the club to their last major trophy - the UEFA Cup in 1997.
He also won the DFB-Pokal in 2001 and 2002, returning to the club for a second spell in charge in September 2011 and leading them to a third-place finish behind Borussia Dortmund and Bayern Munich last season.
With 20 points from the first nine games of the season, Schalke were leading the challenge to Bayern for the title this season.
However, a slump which saw them pick up just five points in the last eight games has seen them fall out of the hunt for the title and out of the top six, which has cost Stevens his job for the second time in Gelsenkirchen.