The France international, who has failed to agree terms on a new deal with the Spanish side, is available for around £18m as he is out of contract in the summer.
He has been offered to a number of English teams but it is understood his representatives are focusing on interest from two clubs in particular, as yet unnamed.
Newcastle have been credited with an interest but Dembele’s wages of around £210,000 a week are considered too much for their structure and the player would ideally join a club competing for European places.
Dembele had a meeting with Barca coach Xavi last week and underlined that he was committed to training and playing despite not agreeing new terms.
However, in a separate meeting with members of the Barcelona board it was made clear that he should continue looking for a new club. their intent was made all the clearer with their move for Wolves winger Adama Traore.
Barca need Dembele off their books before they can register another forward and are hoping the Premier League clubs bite before Monday.