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Having lost 3-2 at home to City in last week's last 16, first-leg in Gelsenkirchen, Schalke want to replace Heidel before the return in Manchester on March 12.
Heidel quit in the wake of Saturday's demoralising 3-0 thrashing of Mainz in the Bundesliga and according to Germany's top-selling daily newspaper Bild, Jochen Schneider, 48, team co-ordinator at RB Leipzig, is the hot favourite to replace him.
Despite having reached the knock-out stages of the Champions League, Schalke are enduring a dire domestic season and the defeat at Mainz was their 12th in 23 league games.
"Even though a lot of things have gone against us this season, I have the overall responsibility for this," said Heidel, who walked away after three years in the job.
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He recruited Domenico Tedesco - the club's 17th head coach since 2001 - for 2017/18 and under the 33-year-old Schalke finished behind Bayern Munich in the Bundesliga last season.
With Heidel gone, Tedesco is no longer safe - unless results improve.
Schalke's chairman Clemens Toennies has said they want to "briskly, but not hastily" find Heidel's replacement.
In addition to Schneider, Michael Reschke, who was recently released as sports director by Stuttgart, has been linked to the role.
Former Belgium head coach Marc Wilmots, ex-Schalke boss Mike Bueskens and ex-Royal Blues midfielder Ingo Anderbruegge, have also been mentioned as possibilities.
The trio were part of the legendary Schalke team, nicknamed 'Eurofighters', which won the UEFA Cup in 1997 after a dramatic semi-final win over Tenerife before beating Inter Milan on penalties in the two-legged final.
Toennies, up for re-election on June 30, is under pressure with the club suffering their worst results since 1983, when they were relegated from Germany's top flight. The team currently are seven points above the bottom three.