The Tricky Trees are determined to bolster their options in the middle of the park, and Barrios has emerged as a potential target.
According to journalist Fabrizio Romano, Forest have submitted a formal proposal of €4m (£3.4m), which was swiftly turned down by the Russian Premier League side.
With Barrios under contract until the summer of 2026, the reigning Russian champions are holding out for a fee in the region of €15m (£12.9m).
Since joining Zenit from Boca Juniors in 2019, the 29-year-old has made 154 competitive appearances, contributing three goals and four assists.
Barrios, who plays predominantly as a defensive midfielder, has also earned 54 caps for Colombia and represented his country at the 2018 World Cup.
Meanwhile, Leeds United have reportedly been made aware of Coventry City's asking price for Dutchman Gustavo Hamer.
Leeds are in the market for a new midfielder after seeing Marc Roca, Adam Forshaw, Weston McKennie depart following the club's relegation from the Premier League.
With Tyler Adams also being linked with a move away, the Yorkshire club have been scouring the market for new additions, with Hamer emerging as a possible option.
According to Football Insider, Leeds have learned that the Sky Blues will accept a bid in excess of £12m for the 26-year-old.
Despite setting an asking price, Coventry are still desperate to retain Hamer's services, as they look to mount a promotion challenge after losing in the playoff finals last season.
Hamer, who is out of contract in June 2024, has also attracted interest from Premier League side Burnley, who had a player-plus-cash bid turned down earlier this summer.
Elsewhere, Aston Villa attacker Philippe Coutinho has reportedly received an approach from an unnamed Qatari club.
Despite being under contract until the summer of 2026, the Brazilian is currently being linked with a move away from Villa Park.
The playmaker has already been approached by Saudi Arabian clubs, with Steven Gerrard's Al Ettifaq believed to be among those showing an interest.
According to journalist Fabrizio Romano, an unnamed Qatari side has entered the race after enquiring about the conditions of a potential deal.
Aston Villa will consider sanctioning a sale if they receive a suitable financial proposal for the former Liverpool and Barcelona man.
Coutinho was unable to nail down a regular place in the side last season, starting just seven of his 20 Premier League appearances.
On the South Coast, Bournemouth's proposed transfer for Fiorentina midfielder Gaetano Castrovilli has fallen through due to a failed medical.
Bournemouth were hoping to make Castrovilli their latest summer addition after recruiting Justin Kluivert, Milos Kerkez, Hamed Traore and Andrei Radu.
After the two clubs reportedly reached an agreement over a £12m transfer, Castrovilli travelled to England on Friday to finalise the move.
However, Fiorentina have since confirmed that Castrovilli has failed a medical with Bournemouth and will return to the Italian club to resume training.
Castrovilli, who was part of the Italian squad that won Euro 2020, underwent knee surgery last year after suffering an anterior cruciate ligament injury.
Despite returning to Fiorentina, there is a possibility that the 26-year-old could still leave the Serie A side before the end of the summer transfer window.