Hearts have been warned they must produce a satisfactory bid for Esmael Goncalves quickly or lose out on the former St Mirren player.
Hearts have failed with two bids for the Anorthosis Famagusta forward, who has worked with both Tynecastle head coach Ian Cathro and assistant Austin MacPhee.
Anorthosis have said they will not stand in the way of the 25-year-old Portuguese player, who helped St Mirren win the League Cup in 2013 while on loan from Cathro's Rio Ave team.
Quoted on Sigma Live, an Anorthosis spokesman told Cyprus's Radio Proto that Goncalves will only be sold if Hearts raise their offer and they secure a replacement.
He added on Thursday: "We expect to finish this matter today or tomorrow. We will not wait for the last days (of the transfer window). "
Anorthosis president Andreas Panteli earlier told Sigma TV: "Their initial proposal was low and we rejected it. They returned with a second bid but that also wasn't satisfactory."
The Cypriot side have already identified a potential replacement in AEK Larnaca forward Andre Alves.
Cathro refused to expand on his interest in Goncalves after his side's 4-2 extra-time win over Raith Rovers on Wednesday night, which set up a William Hill Scottish Cup fifth-round tie with holders Hibernian.
But he revealed Hearts had a number of targets.
"We are not at a point to be able to identify anybody specific," he said. "We definitely do want to do some business in that area of the pitch and that type of profile of striker but we have a number that we will be looking into and we need to be correct in what we do over the next few days and hopefully bring one of them to a close."
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