Fellow Lion Leigh Halfpenny has firmly established himself as his country's first-choice 15, with Byrne struggling even to make the wider Wales squad.
The 32-year-old's decision to swap the PRO12 for the Top 14 has left him outside of the loop but his fine form for Clermont Auvergne finally saw him recalled to the Six Nations squad after a 15-month absence.
Byrne missed out on selection for the tournament opener, but Jones, who coached Byrne at the Liberty Stadium, believes he should be given his 47th cap - and his first since the 2011 World Cup - when Wales travel to Paris this Saturday.
"Leigh Halfpenny's got to play because of his goalkicking but I would certainly find a place for Lee Byrne in the side," Jones told Wales Online.
"He is a little bit more of an attacking threat out wide and an experienced campaigner who hits some nice angles from full-back, so that's one change I would certainly make."
Byrne's current club coach Vern Cotter said last month that he was surprised the former Llanelli player hadn't been recalled by Wales far earlier, citing his impressive performances in both attack and defence for Clermont since his move to France 18 months ago.
Cotter even stated that he 'wouldn't take him out of Lions contention if he gets a couple of games', with Byrne's supporters now hoping that this weekend will finally see him back in a matchday 23 for Wales as the clock ticks down towards selection for Britain and Ireland's elite at the end of April.
Byrne isn't the only name being linked with a promotion to the Welsh side after last weekend's defeat to Ireland, though. Interim head coach Rob Howley is under pressure to name a number of new faces in his XV after Wales fell behind 30-3 in a frustrating first 43 minutes at the Millennium Stadium.
Skipper Sam Warburton and Lions front rowers Gethin Jenkins and Matthew Rees are among those facing a battle to keep their starting spots, with the likes of Justin Tipuric, Paul James, Richard Hibbard and Ryan Jones all pressing their claims for inclusion.
"When you don't play as well as you can play you come under pressure, because there are good individuals underneath you who want that jersey," said Wales assistant coach and former Test wing Mark Jones.
"Pressure is part of this sport, part of your job. Player selection is down to form and injury. Ours, for the first game, was predominantly around who we had available.
"Do we give the players an opportunity to go out and redeem themselves or do we try and freshen things up and bring some fresh blood in? Those are the type of things coaches have got to make decisions on. They are not easy ones, to be honest with you."
Wales are set to name their side to face France on Thursday.