The Mexican still has a year remaining on his Firhill contract but could leave the Jags in the coming weeks after falling down the pecking order.
SportTimes understands Piccolo sought assurances over his place in Alan Archibald's plans several weeks ago and while he is keen to remain at Firhill, he is not guaranteed regular first team football next term and could therefore look to move on this month.
Boss Archibald remains in the market for a left-back and centre-back this summer following the departures of Aaron Taylor-Sinclair and Lee Mair and Piccolo could become the third defender to leave the club if the right offer comes along.
SportTimes can reveal that clubs in Portugal, Cyprus and Greece have expressed an interest in Piccolo but the 24-year-old is eager to prove his worth at Firhill and could yet stay and fight for his place ahead of the new Premiership campaign.
Piccolo made 21 appearances for Thistle last season as the Jags retained their top flight status but with Conrad Balatoni and Aaron Muirhead already ahead of him in Archibald's thinking and more new arrivals likely, it is understood he will be free to leave Firhill if he wishes.
The Thistle boss cast his eye over another left-back contract hopeful in the bounce game with Rangers this week and is continuing his search for at least three more new faces having already snapped up Ryan Stevenson and Abdul Osman ahead of the big kick-off.
Meanwhile, Thistle's Under-17 side saw their dreams of Milk Cup success come to an end last night as they were defeated by Manchester United at the Coleraine Showgrounds.
The Jags have shone across the water in recent days but their memorable run was halted in a 4-0 defeat by the Red Devils at the semi-final stage.