Jones, 23, is contracted to Cape Town-based Western Province and Stormers franchise until October.
He will remain in South Africa to compete in the Super Rugby competition, which ends in August.
If a deal is done between Glasgow and the player, he could move to Scotstoun when the Stormers' involvement in the tournament comes to an end.
Discussions are also ongoing between Glasgow and Scotland under-20 fly-half Adam Hastings over a summer move from Bath.
Hastings, 20, is the son of Scotland great Gavin Hastings, and has made five appearances for the Aviva Premiership club this season.
Scottish Rugby has shown interest in recruiting Jones, who has been named in head coach Vern Cotter's Six Nations squad, for several months.
Glasgow have tracked his development since he enrolled at University of Cape Town in 2014.
Jones scored twice in the autumn loss to Australia and created Sean Maitland's try in the win over Argentina a week later.
Fellow Scotland centres Alex Dunbar and Mark Bennett are two of the bigger names whose Scotstoun deals expire at the end of the season, and have yet to re-sign.
Australian centre Sam Johnson has emerged as a key element of Glasgow's midfield this season, while Richie Vernon is set to return from injury after undergoing shoulder surgery in November. However, both their deals also expire in the summer.
In the case of Hastings, who is behind England's George Ford and Rhys Priestland of Wales in the battle for the Bath number 10 jersey, a move north would present the highly-rated pivot with the opportunity to play more first-team rugby.
Scotland's Finn Russell is the only fly-half contracted to Glasgow beyond the end of the season.
Glasgow's versatile Peter Horne can also play at fly-half, but is recovering from an ankle injury that could rule him out until late March.  Although his contract also expires this summer, he is expected to sign a new deal at the club.
Glasgow Warriors are to offer a contract to Scotland centre Huw Jones.