Hungarian international stopper Fulop has been on loan at the Walkers Stadium since the start of the season and has become a firm favourite with the fans over the course of 26 games in which he has kept nine clean sheets.
And City boss Ian Holloway is determined to show supporters at his new club that he wants to make the 6ft 5in keeper part of his rebuilding plans.
Fulop originally went to the Midlands club on a season-long loan, but Sunderland had an agreement in the deal allowing them to recall the keeper in January if they wanted.
And by activating that clause, as they have done, they have forced Leicester to make a decision on buying the 24-year-old.
Sunderland boss Roy Keane is anxious to strengthen in the transfer window and will be given funds by the board to do so, but the consortium are also anxious that he raises money from within the club by offloading anyone who is not a real part of the manager's long-term plans.
Keane told the Echo last week that he was happy to have Fulop back at the club and was a believer in big clubs needing four keepers.
But the £1million fee would bolster his kitty and he has said previously that he has every confidence in Craig Gordon, Darren Ward and promising youngster Trevor Carson.
Fulop was signed by Sunderland from Tottenham in the last January transfer window, as part of the deal which took fellow keeper Ben Alnwick in the opposite direction; Fulop costing £500,000.
And it might not be a bad bit of business for Sunderland to have doubled their money in the space of 12 months on a fringe player.
We will have to wait and see whether Leicester's £1million bid is acceptable, whether or not City will be asked to come up with a bigger offer or whether another club will come in for the giant keeper's signature.
Meanwhile, City find themselves desperate for an experienced, quality midfielder and are looking at Kavanagh (pictured).
The midfielder was signed on generous wages last season as Sunderland tempted him to step down from the Premier League.
And that has been the problem for cash-starved Championship teams in terms of him leaving in the January transfer window.
First Sheffield Wednesday and then Norwich City found there was no way they could put a financial package together which would not have cost the 34-year-old a small fortune in lost wages.
But Leicester have inquired and are hoping to broker some sort of deal for a player who looked a class act for Sunderland in the Championship last season.