The veteran Frenchman is out of contract at the end of the season, which has alerted the struggling duo.
Despite this, Distin is believed to be willing to stay at Goodison Park and fight for his future as the club look to maintain their push for European qualification.
Everton boss David Moyes is thought to be looking to strengthen his defence, and to do so he must reduce number in his squad.
However, he is said to value the 35-year-old highly and could offer him a new short-term deal at the end of the current campaign.
Both QPR and Sunderland are willing to hand him a two-year contract and pay a fee to Everton in order to push through a January move.
Harry Redknapp is openly courting defensive options with his side sitting bottom of the table, whilst Black Cats manager Martin O’Neill is keen to add experience and strength to his rear-guard.
But, Distin is thought to prefer the prospect of staying on Merseyside, where he has been since 2009, and would likely reject any move within the coming weeks.
The centre-back has a wealth of Premier League know how, having played for Manchester City, Portsmouth and Newcastle before joining Everton.