The 28-year-old is one of Celtic's most influential players, netting 17 goals in all competitions last season to win Player of the Year.
The report adds that the prospect of losing Forrest is 'unthinkable' for manager Neil Lennon following Tierney's £25 million move, and that the player himself could consider a move abroad for financial gain.
The Russians, who have just dropped £40 million on signing Malcom, certainly have the funds to be a threat to the Scots.
Forrest is contracted to the Parkhead club until 2022 but Lennon has revealed recently that he hopes to sign him up to an extension.
Tosh McKinlay compiled a report on the left-back as Taylor’s Kilmarnock side went down 2-0 at Hamilton on Saturday. Record Sport exclusively revealed last week the 21-year-old was among several names under consideration to replace Tierney, who completed his £25million move to Arsenal last Thursday.
Celtic manager Lennon is poised to hold talks with chief executive Peter Lawwell and his recruitment department this week as they look to strengthen their squad for the domestic and what they hope will be a prolonged Champions League assaults.
Former Aberdeen loanee Max Lowe is also mentioned in the Record report with the 22-year-old playing in just 35 minutes of Derby’s two opening Championship fixtures.
The full-back spent last season on loan at the Dons where he made 42 appearances.
Celtic have now also entered the race for Lowe after holding talks with the Championship side.
The Hoops are extremely short of defenders area after the departures of Emilio Izaguirre, Dedryck Boyata, Filip Benkovic and Mikael Lustig.
Celtic are believed to be chasing Brentford's 7m-rated left-back Rico Henry, as they look to secure a replacement for Kieran Tierney. (Scottish Sun)
The report has also claimed that Brentford value the 22-year-old left-back at £7 million, but the Championship club will offer the former England Under-20 international a new contract.
The 22-year-old is a very good and promising young left-back, and at the reported price of £7m, the Englishman would be a smart signing for the Hoops.
According to WhoScored, the former England Under-20 international made 13 starts and one substitute appearances in the Championship for Brentford last season, and has played twice in the league for the Bees this campaign.
Writing in The Record, Chris Sutton has called on the Celtic board to make 'serious money' available to Neil Lennon between now and the end of the summer transfer window.
There are also some big-name players linked with a move away from Glasgow and, according to The Record, Lille are set to make an improved bid for Jozo Simunovic after a £4 million offer was recently knocked back.
And Sutton claims that the Northern Irishman should walk away from Celtic if £5 million is accepted for the big Croat.
He told The Record: "There’s no getting away from the fact Celtic need new faces though. If Lennon is not given serious money to do it then it will be nothing short of negligent and anyone on the board who thinks the squad has enough right now is delusional.
"You can’t have the manager working with one hand behind his back. There’s talk of interest in Jozo Simunovic but if Celtic accept £4m or £5m then Lennon would be as well walking away."
Simunovic has established himself as a very big player for the Bhoys since Lennon took the reins in February and losing him would undoubtedly be a blow.
Emilio Izaguirre has killed off any lingering speculation that he could be on his way to a third stint at Celtic.
After three months trying to get a deal sorted out the defender has signed for Honduran side Motagua within a week of Kieran Tierney joining Arsenal in a £25m deal.
Izaguirre made 16 appearances for Celtic last season after a stint in Saudi Arabia but a third return became a possibility on July 27 when the Daily Record reported Montagua president Eddy Atala saying:
We spoke to Emilio Izaguirre’s agents this week and he has offers from clubs in the USA, Japan and Australia. There is also the possibility of him returning to Celtic, but that is only for a short time of six months.
Celtic are reportedly interested in signing Yannick Bolasie from Everton. According to a report by Goal, Everton are hoping to offload Bolasie soon with three clubs chasing the winger.
Bolasie turned down the chance to move to CSKA Moscow recently but he has other options on the table. Everton failed to move Bolasie on during the English transfer window so he has to move to another league before September 2.
Three clubs want Bolasie
Celtic are one of the three clubs trying to snap the DR Congo international up. The Bhoys face competition from French club Rennes and Turkish heavyweights Besiktas in this transfer tussle.
Bolasie is out of favour at Everton and will be allowed to leave. The 30-year-old spent last season on loan at Aston Villa and Anderlecht as Marco Silva brought in other options at Everton.
Currently, Bolasie is behind Richarlison, Gylfi Sigurdsson, Bernard, Theo Walcott and new signing Alex Iwobi in the pecking order. This makes a move likely with Celtic hoping to sign him.
Celtic have won both of their league games this season and are in the process of Champions League qualification. Bolasie could bolster their title and European credentials. Here are some fans who are excited by this rumour.