Press Association Sport understands a club in the Gulf state have tabled an offer for the 26-year-old Ghana international, who is currently on a season-long loan deal in the United Arab Emirates with Al-Ain.
Gyan will have two years of the contract he signed when he joined the Black Cats from Rennes during the summer of 2010 remaining at the end the current season, and his future could have a major bearing on manager Martin O'Neill's transfer plans.
The club's identity has not been disclosed.
Gyan arrived on Wearside on the back of a series of impressive displays for his country at the 2010 World Cup finals in South Africa as then manager Steve Bruce invested a club record £13.1million in him.
It took Gyan time to get used to the Barclays Premier League, but he announced his arrival in earnest with is first goal for the club at Wigan on September 11.
He managed 11 goals last season as he, Darren Bent and Danny Welbeck vied for places in the starting line-up before Bent's January departure for Aston Villa, and also scored for Ghana in a friendly against England.
He ended the campaign as the club's only fit striker as injuries to Welbeck and Fraizer Campbell took their toll on Bruce's squad.
Gyan was linked with a summer move to Tottenham as Bruce renewed his interest in former target Peter Crouch, and despite the manager's repeated insistence that the African was going nowhere, the noises coming out of player's camp were not encouraging.
It nevertheless came as something of a shock when, hours before the club's home clash with Chelsea on September 10, they announced that Gyan had joined Al-Ain on loan for a fee of around £6million.
The terms of the deal meant he could not be recalled when Martin O'Neill replaced Bruce as manager in December.
Sunderland are understood to have sought talks over his future plans as they attempt to find a way forward amid speculation that Al-Ain want a permanent deal, and that there are other interested parties.