Chelsea and Zenit's valuations for Leandro Paredes are currently €10 million (£9m/$11m) apart after the Blues were asked to increase their offer for the midfielder to €40m (£36m/$46m).
Goal understands the Premier League club's initial approach for Paredes was €30m (£27m/$34m), and the closeness of the two valuations means that there could soon be a positive outcome for the Argentina international.
Paredes, who showed up to conduct medical tests on Monday with Zenit after a winter break period in the Russian Premier League, is seen by Chelsea boss Maurizio Sarri as an ideal type of player to come in and deputise for Jorginho.
Sarri is keen to secure a replacement for Cesc Fabregas, who was allowed to complete a free transfer to Monaco on Friday, after a deal was initially agreed earlier in the month .
Fabregas made his debut for Monaco in Sunday's 1-1 draw against Marseille and performance-related clauses mean that Chelsea could earn £10m ($13m) over the course of his three-and-a-half year contract.
Chelsea beat Newcastle 2-1 on Saturday, but Jorginho struggled to meet expectations at Stamford Bridge, which Sarri admitted after the game.
The Chelsea boss also said that he would have substituted Jorginho had Fabregas been involved and described his lack of options for the central midfield pivot as the one weakness currently in the squad.
“If Fabregas was on the bench, I think that in the second half then Fabregas would be on the pitch instead of Jorginho," Sarri said on Saturday.
“In the future maybe [we can try Ethan Ampadu there], but we have to work. Playing as a central midfielder in our team is not easy, we need to try and try and try in training with him.
“The player who plays in that position usually touches the ball about 120-130 times and so I think that we need a specialist there. I think that he is able to do it.”
Chelsea have other situations going on in the January transfer window, as striker Alvaro Morata could be replaced by Gonzalo Higuain but ongoing talks remain complicated.
The future of Callum Hudson-Odoi is in doubt after a £35m ($45m) offer from Bayern Munich turned the 18-year-old's head, but Chelsea's hopes of convincing the youngster to stay have led to a stalemate that could be played out over many months.