I would guess that it depends a lot on the club whether they would write that sort of thing into players’ contracts. It’s hard to imagine the contract that Wayne Rooney just signed at ManU (rumoured to be in the region of 300K per week) having a relgation clause in it. OTOH, clubs that are used to bouncing between the divisions, or recognise they are a relegation risk, might well do something like that.
Man City were relegated twice about 15 years ago, to the third tier of English football - unprecedented in the modern era for a club of that size and history. They got out of the third tier by the absolute skin of their teeth - they were dead and buried in a play-off game and staged a Lazarus-like comeback. Had they not done that the club would have been at serious risk of going under, from carrying too many permiership contracts down the divisions. They are now the wealthiest club in the world, which is an interesting change of circumstance.
There are examples of small clubs who made it to the premiership and rashly splashed out on premiership contracts for players, which turned into lead anchors as the club fell back down the divisions. Barnsley springs to mind - think they had a period of administration following successive relegations.