West Ham have lodged a third bid – thought to be around £30m – for Burnley defender James Tarkowski.
The Hammers had a second bid of £27m rejected earlier this week.
West Ham boss David Moyes is keen to bolster his defence as the club prepare to start their Premier League season at home to Newcastle on Saturday.
Tarkowski, 27, joined Burnley from Brentford in 2016, helping his team finish 10th in the Premier League last season.
He has two England caps and is regarded by many as being unlucky to miss out on selection in Gareth Southgate’s squad for the recent Nations League qualifiers against Iceland and Denmark.
West Ham are under pressure from fans after the sale of homegrown attacking midfielder Grady Diangana to West Brom, a decision that was criticised publicly by captain Mark Noble.
It is understood the sale was approved reluctantly by the Hammers. However, after owner David Sullivan injected money into the club to ensure bills were paid during the coronavirus lockdown, Moyes was told he would need to sell players before he could buy new ones.
Hoped-for deals involving Felipe Anderson and Manuel Lanzini failed to materialise, meaning West Brom’s offer for Diangana had to be accepted if Moyes was to be able to make signings he feels are needed to avoid a repeat of last season’s relegation battle.