QPR are already said to have had three bids rejected for the 27-year-old US international.
Ream, 27, has established himself as one of Bolton’s most reliable performers since arriving from the New York Red Bulls in 2012 and won his second straight Player of the Year award last season after helping Neil Lennon’s side escape relegation trouble.
Making 44 league appearances, his impressive form earned him a spot in the USA squad for last month’s CONCACAF Gold Cup, where he started at left-back in both the Group A win over Haiti and third-place match against Panama, and has also caused interest in his signature heat up in recent weeks.
MLSsoccer.com’s report claims that QPR have so far had three bids rejected for the US international but are believed to be preparing a fourth, having seen Bolton boss Neil Lennon label their latest offer of around £1 million as well below his valuation.
"I don’t want to sell him and the club doesn’t want to sell him,” Lennon told the Bolton News last week. "It’s important we look to build a squad and Tim is part of that.
"He played a lot of games last year and was very consistent. I think he will get better as well. He is important to what we want to do here and there is no real interest in selling.”
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Nevertheless, with finances tight at the Macron Stadium, Lennon has also admitted that the right offer could be enough to secure the 13-times capped American’s signature.
"Look, at the end of the day every player has his price,” he added. "I’m not saying we will refuse offers, so long as it suits us but at the minute it doesn’t.”
QPR now look set to step up their pursuit of Ream during the final weeks of the summer transfer window but, if the latest claims are to be believed, they could well be about to face some strong competition from London neighbours Fulham.
Compatriots Carlos Bocanegra, Clint Dempsey, Marcus Hahnemann, Eddie Johnson, Kasey Keller, Eddie Lewis, Brian McBride and current midfielder Emerson Hyndman have all turned out for the club dubbed “Fulhamerica” over past seasons, and now the St Louis-native is said to be the next American on their radar.
Able to operate at centre-half, left-back or even in midfield, his age, Championship experience and versatility would certainly make him a solid addition for any promotion pushing side, but it remains to be seen whether Bolton will receive an enticing enough offer between now and transfer window closing.