The 47-year-old is still under contract until 2020, but the London Evening Standard suggests that he will be given the chance to double his £2.5m yearly salary when discussions begin over fresh terms in the coming weeks.
Loftus-Cheek has been impressed by Southgate's management ability and wants the ex-Middlesbrough chief to continue beyond the next World Cup in Qatar.
"It will be great news if he signs another deal and is around for more tournaments," he told reporters. "We have such a young team, full of potential.
"There are players out there who didn't quite make the World Cup squad as well, so to have him around for many more years will be a great thing for everybody.
"He is a top manager and knows what it's like to be a player. We all have such massive respect for him. Him being here for the long haul is great news for us and for England."
If he pens a new deal, Southgate will be on course to be England's longest-serving manager since Sven-Goran Eriksson between 2001 and 2006.