The Iceland skipper was carried off during his sides 4 - 2 defeat at home to Slovenia on the weekend and now faces a race to be fit for the start of next season.
"We will give Aron the best possible support to enable a swift return to action," said Head of Performance Richard Collinge.
"Aron is currently undergoing a series of tests and investigations to determine the extent of his shoulder injury, from which we will be able to implement a management and recovery schedule."
But the player himself has given a positive message. He tweeted on Monday
'Thanks for all the messages folks, i am okey and will be alright'
Gunnarsson was x-rayed in Iceland and is now back in South Wales.
His international boss Lars Lagerback said
“He had an X-ray in hospital which should reveal how serious the injury is.
“For us, it had a big impact on the course of the game.
“We were level at 2-2 before our captain went off.
“We were unlucky that Aron injured himself because the balance of the game changed when he went off.
“Hopefully Aron will recover soon as he will go on holiday now and has a few weeks to recover before the Premier League season starts.”