Lee Miller has been in excellent form for Hearts
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Bristol City are asking Hearts to make them a final offer for Lee Miller in writing if they are serious about signing the 21-year-old striker.
Talks have taken place and verbal offers have been made and rejected for Miller, who has been on loan at Tynecastle since January.
City want £400,000 for the striker, who has scored 11 goals for Hearts.
City chairman Steve Lansdown last week cast doubt on whether the sale would now be completed.
"As things stand, I am not sure any permanent deal will go through," Lansdown told his club's website.
"I'm really disappointed with the current situation."
Hearts have offered £200,000 plus incentives that could match the £325,000 City paid Falkirk.
And the Scottish Premier League club are said to be unhappy that City have increased their demands since January, when the asking price was £175,000.
Hearts could not meet the asking price at that time as they were in the middle of a takeover and instead opted to take the 21-year-old on loan.
"At the beginning of this week, I felt we were close to a deal," said Lansdown.
"The things we had spoken of appeared agreeable to both sides.
"Now there seems to have been a hiccup at the Hearts end and I'm not too sure what it is about."
Miller, bought by City from the Bairns in 2003, whose 11 goals have come in 19 games for Hearts.
"People questioned why we paid £325,000 for him, but we felt there was potential there," said Lansdown.
"If they do not meet the asking price, he will return to us a better player. He's also attracting other interest."
Miller has another year of his contract to run at Ashton Gate.
Meanwhile, more players are expected at Tynecastle for trails before the end of the season.
Former Sampdoria goalkeeper Marco Fantini is one of a number of players to have already trained with Hearts.
The 20-year-old played for Genoa club last season, but his football career was interrupted by a year's compulsory military service.