Chelsea offered Zenit St Petersburg an opening of £ 26.8 for midfielder Leandro Paredes.
After Cesc Fabregas left for Monaco on Friday, Paredes was probably the most likely arrival to move west of London.
Coach Maurizio Sarri made it clear to the hierarchy of Chelsea that he wanted Fabregas to be replaced and the club also looked at Cagliari star Nicolo Barella.
The latter, however, has reservations about moving to the capital this month and is also subject to considerable interest from Napoli and Inter Milan during a summer move to Serie A.
Paredes, who won nine caps for Argentina, reported that he wanted to leave Zenit to return to the Boca Juniors Boyhood Club.
However, Chelsea is prepared to change his mind, as they are ready to give him a contract of four and a half years with a basic salary of around £ 80,000 per week.
Zenit could still be convincing to sell at this price, as they paid Roma £ 24 million for Paredes in 2017, which included various add- ons.
But Chelsea is under pressure to strengthen its position and, if necessary, will pay more.
Jorginho has very little backup and Sarri admitted that he would have replaced the Italian in Saturday’s 2-1 win over Newcastle if Fabregas had been on the bench because he was fighting so much.