The 35-year-old currently sits at 75 caps but last played for his country in 2017. He was left out of Gareth Southgate's squad for the 2018 World Cup.
Towards the end of his time at Manchester City Hart had loan spells with Torino and West Ham United before moving to Burnley.
He started his spell at Turf Moor as No.1 but was dropped to the bench at the end of 2018. Between 2019 and joining Celtic last summer, Hart made just 13 appearances and didn't feature in the Premier League.
In Glasgow he has rejuvenated his career, playing a key role as Celtic reclaimed the Scottish Premiership title last season, while being an important figure off the park.
Aaron Ramsadle, Jordan Pickford and Nick Pope were the last three goalkeepers selected by Southgate but Sutton reckons there could be a place for Hart at the World Cup in Qatar later this year.
"I’m not in the dressing room, but from what I can tell, Hart has been absolutely vital to the success of the team as he’s a leader in there and he will go back into the Champions League group stage this season as one of the club’s most influential figures," he wrote in his Daily Record column.
"Will he get to Qatar? Chances are no, even if he shines in Europe.
"Jordan Pickford is doing well and has never let Gareth Southgate down, while Aaron Ramsdale is also performing well at Arsenal. The No.3 slot might be up for grabs come November and there could be injuries and you wouldn’t put it past Hart shining in the Champions League and stating a World Cup case.”
Earlier this year, Celtic boss Ange Postecoglou paid tribute to Hart and his influence.
"I thought Joe was a top- quality keeper, but the background we did on him showed me he is a first-class person and a real leader in the dressing room," he said.
“I just felt he’s still got a bit to prove. He’s said himself, he needed a bit of love and I knew he’d get it at this club.”