Grealish, 30, has been expected to leave City this summer after falling down the pecking order, with Everton the most likely destination following a productive loan spell at Goodison Park last season.
That move was cut short by a foot injury in January, which also ended his hopes of a World Cup call-up with England.
According to the Daily Mail, City would be open to a compromise on a deal, but there is now concern on Everton's side, with the club reportedly questioning Grealish's professionalism before pushing ahead.
Before his injury, Grealish had made 20 Premier League appearances for Everton, scoring twice and adding six assists as the club pushed for a European spot. Everton's form dipped after his injury and they eventually finished 13th, a slide that has fed into the club's current caution over committing to a permanent deal.
Grealish is in the final 12 months of his £300,000-a-week contract at City, having been left out of the club's pre-season tour squad as he continued his recovery. His return to first-team training this week, without any resolution on his future, has only added to the uncertainty surrounding where he will be playing his football when the new season begins.
Manager Enzo Maresca left the door open to a return to the first-team fold last month.
"At the moment, he is here. He is a Manchester City player," Maresca said. "I always said the same. In any club I join, I like to know if there are players that belong to the club. It's my duty to try to coach them. Jack is here.
"I have a good relationship with Jack since I left. Also, we keep in touch, and the reason why is because he has a big heart and he is a very good guy. So this is the reason why and then we see what happens."