The 27-year-old is transfer listed after one season at the club and chairman Darragh MacAnthony has confirmed that two bids have been received this summer but don’t meet his asking price.
Hearts are looking to bolster their squad in all areas for a demanding season, with a minimum of eight European games before Christmas.
Grant featured in 26 of Peterborough’s Championship games last term, scoring two goals and providing three assists, but they were relegated to league One. The former Nottingham Forest youngster has been with Peterborough since last summer when they brought him in from Lincoln City.
He has been on Hearts’ radar and a Football Scotland report has linked him with a potential move, but the Edinburgh Evening News understands the Tynecastle club are not prepared to pay anything like the £750,000 fee Peterborough are asking for.
The Peterborough Telegraph described his first season at the Posh as a “huge disappointment” following his move from Lincoln. However, he had been a big influence at Lincoln in 2020/2021, scoring 17 goals and being named in the League One team of the year.
Posh triggered a release clause in Grant’s contract to sign him and director of football Barry Fry has revealed he has a similar clause in his current deal, although not one that is likely to be met by Hearts or other potential suitors.
"We got Jorge on the cheap considering how good he had been at Lincoln,” Fry told the Peterborough Telegraph. “He also has a release clause in his Posh contract in the event of relegation from the Championship, but I doubt anyone would reach it.
"He did struggle last season and it’s difficult to see him fitting into the manager’s style of play, but we do want a fee for him. We want to move him on, but if he stays he was at least very good the last time he played in League One.”
Grant also has stints at Mansfield, Luton, Notts County and Nottingham Forest on his CV.
His best spell was at Lincoln, for whom he made 88 appearances, scoring 19 goals and collecting 24 assists. He also has 25 goals and 12 assists from 73 appearances for Notts County.