Hegarty has forced his way into the Gers first team this season, along with the likes of Robbie Crawford, Barrie McKay and Lewis Macleod.
He has been involved in 14 games for Ally McCoist's side and has helped them storm 19 points clear at the top of the Irn-Bru Third Division.
The centre-half, who has often been played out of position at right-back, is out of contract at the end of this campaign.
But Rangers manager McCoist is keen to keep the hugely promising Northern Ireland Under-21 captain in Glasgow.
Youth graduates Crawford, McKay and Macleod all signed five-year deals last September that will keep them at Ibrox until 2017.
And first-team coach Ian Durrant confirmed that the 20-year-old from Dungannon is on the brink of pledging himself to his boyhood heroes.
Durrant said: "If the boys are doing their jobs then they will get new contracts. Young Chris is on the verge of signing an extension."
McCoist, who can register free agents as players on September 1 after a 12-month signing ban ends, is speaking to targets at the moment.
Durrant is hopeful that, as well as retaining the services of existing players like Hegarty, Rangers can strengthen their squad further.
And he revealed how the huge attendances Rangers have been getting at Ibrox this season – despite playing in the bottom division – is helping their recruitment drive.
He said: "Barrie McKay, Lewis Macleod and Robbie Crawford are all tied down. The likes of Ian Black and Dean Shiels are here for three or four years.
"We have a good block of players here. Having said that, they will need a hand. If we can get fresh blood in it will help them.
"Hopefully, if we can make a couple of top signings, then we will be able to push on again and improve further next season."