Upson had a loan spell with Barnet last season, scoring one goal in nine matches for the League Two side.
The Glovers have also had talks with ex-Bristol City and Swansea striker Lee Trundle, who was not offered new terms at Ashton Gate.
"We have spoken," Yeovil manager Terry Skiverton told BBC Radio Bristol. "He's a Championship-type player and two years ago he was worth £1m.
"We've made him an offer and he's mulling it over but there will be Championship clubs out there who will be interested in him.
"We'll just have to wait and see."
Yeovil reported back for pre-season training boasting four new signings, with Luke Ayling and Luke Freeman joining Upson and former Bristol Rovers winger Andy Williams.
And Skiverton hinted there will be more to come before the season kicks off on 7 August.
"We're doing things the right way and our reputation is starting to go out there," he added.
"We're trying to build the club for the future. We've tried to strike early with a couple of young players and get them for the foreseeable future.
"Last year was a baptism of fire for me, this year I'm trying to move the club forward so we'll be in a lot stronger position.
"I'll be bringing in a little bit of experience to help these youngsters - it's going to be exciting."
Source: BBC Sport
Source: BBC Sport