Fifa president Gianni Infantino invited the team to the World Cup final in Moscow on Sunday.
Although the boys are too ill to attend, Thai football officials will meet Fifa in Moscow this weekend to explore inviting the boys to a future event.
Manchester United were among the first of football’s giants to invite the team to visit their stadium.
In a statement, the club said: “We would love to welcome the team from Wild Boars Football Club and their rescuers to Old Trafford this coming season.”
Italian side Roma wrote: “The best football news of the summer – ALL 12 young Thai players and their coach have been rescued.”
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Portuguese club Benfica has offered the Thai boys an all expenses-paid week at its training academy.
Club president Luis Filipe Vieira wrote to the Thai ambassador in Portugal to invite the boys to the camp in Seixal, near Lisbon.
He said: “We believe this simple gesture can help these faces recover the joy and smiles that no child should ever lose,”
Manchester City defender Kyle Walker, who is starring for England at the World Cup, offered to send shirts to the team.
He tweeted: “Amazing news that all of the Thai kids are out of the cave safely!
“I’d like to send out shirts to them! Is there anyone who can help with an address?”