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The cash-strapped Catalans would prefer to agree a loan deal for the Sterling, but it is thought that City would prefer to strike a permanent deal for the England winger next summer and not see their squad reduced halfway through the season.
Si.com reported how Toni Juanmartí relayed a tweet by Reshad Rahman, which stated that Barcelona are continuing to "maintain their interest" in Sterling, with the player himself open to a move to the Camp Nou.
The 26-year-old has started just three of City's Premier League games this season, in what has so far been an underwhelming start to the campaign.
He has managed just 1 goal in 14 appearances across all competitions, which came in a 5-0 win over Norwich City back in August.
It is big fall from grace for Sterling, who was one of England's brightest stars in their run to the European Championships Final, scoring three goals in the games against Croatia, the Czech Republic and Germany.
His contract at the Etihad expires in June 2023 and Pep Guardiola will want to make a decision on Sterling's future as soon as possible, especially if a new deal cannot be agreed.
Any permanent transfer is likely to be costly, with Sterling currently the sixth-most valuable player in the Premier League and priced at £81million according to transfermarkt.