The News understands the midfielder has attracted interest from Pompey's League One rivals as he edges towards a Fratton Park departure.
Close has been offered fresh terms to extend his stay at his boyhood club.
However, his new contract would be on reduced terms.
After spending the majority of the season in the cold under former manager Kenny Jackett, Close was brought firmly back into the fold after Danny Cowley took over the PO4 hot seat in March.
The academy product started 10 of Cowley's 12 games in charge as Pompey narrowly missed out on the play-offs.
In total, Close has made 190 appearances for the Blues, scoring 19 goals.
He helped Pompey claim the Checkatrade Trophy at Wembley in 2019 but now looks set to exit the south coast.
The 24-year-old has been exploring his options as a free agent this summer.
And it is understood Doncaster are keen on a move.
Rovers have recently appointed Richie Wellens as manager ahead of the 2021-22 season.
And the ex-Swindon manager is looking to prise Close to the Keepmoat as he aims to build on Donny’s disappointing 14th-place finish.
What's more, Noel Hunt has arrived as Wellens' right-hand man in Yorkshire.
The Irishman's someone who knows Close well, having shared a dressing room together at Pompey during the 2016-17 League Two title-winning campaign.