Gelhardt signed a four-year contract with the Premier League club back in 2020 after making the switch from Wigan Athletic.
The England U20 international made 23 appearances for Leeds last season, including 20 in the Premier League.
According to Football Insider, the West Yorkshire club are in advanced talks with the 20-year-old over a new deal.
The report also suggests that Gelhardt could be in line for a big pay rise alongside the new contract.
In less positive news for the club, there are concerns over the future of Jack Harrison who are linked with both Tottenham and Newcastle.
A reported fee of £30million initially starved off interest from the North London club, however, one team willing to spend that kind of cash are the Magpies.
Leeds have already lost Kalvin Phillips and Raphinha who moved to Manchester City and Barcelona.
Losing a third first team player would be a worry for Jesse Marsch who is already in the rebuilding process.
Harrison made 165 appearances for Leeds, including his period at Elland Road on loan from Manchester City.
The 25-year-old has also contributed with 28 goals, 10 of which came last season.
Leeds signed Harrison on a permanent deal in 2021 for a reported fee of £11million. The player signed a three-year contract, keeping him at Elland Road until 2024.