Vieira, 26, spent last season on loan at newly-promoted Hamburg, where he was one of the team's standout performers. The midfielder scored seven goals and provided five assists in 29 Bundesliga appearances during his loan spell.
Those performances significantly boosted his market value following a difficult few years at Arsenal, where he struggled to establish himself after joining from Porto. Hamburg held a purchase option for Vieira worth around €22m, but allowed it to expire after deciding the fee was too high.
The Bundesliga club had since been in discussions with Arsenal over a reduced fee, with earlier reports in Germany suggesting figures in the region of €10m to €15m were under discussion.
According to journalist Dominik Schneider, Nottingham Forest have now submitted a formal offer worth €28m in total for the midfielder.
The bid is reported to have come as a surprise to those involved in Hamburg's negotiations, significantly strengthening Arsenal's position. Sky Sports' Florian Plettenberg has since played down the reports, saying there are currently no active negotiations between the parties involved. According to Plettenberg, all sides have denied that formal talks have taken place, despite the reported figure being widely circulated in German media.
Vieira has just one year remaining on his contract at Arsenal, a factor that had previously been cited by Hamburg as part of their case for a lower fee. Since Vieira left on loan, Arsenal have strengthened their central midfield options significantly, with the additions of Martin Zubimendi and Eberechi Eze contributing to the club's Premier League title win.
Academy graduate Myles Lewis-Skelly has also added to Arsenal's options after impressing in a midfield role during the closing stages of last season. That depth means Vieira's path back into Mikel Arteta's first-team plans appears limited, regardless of which club he ultimately joins.
For Arsenal, the emergence of Nottingham Forest's interest, even if not yet at the negotiating stage, gives the club additional leverage in any discussions with Hamburg. Should Forest's reported figure prove genuine, it would represent a stronger return for Arsenal than the fee currently being discussed with the German club. Vieira previously spent time on loan at Porto before his move to Hamburg, with his performances in the Bundesliga having revived his prospects considerably.
For Hamburg, losing out on Vieira would represent a blow after building much of their promotion-winning campaign around his creativity in midfield.
The coming days are likely to clarify matters, with Arsenal now holding multiple options as they decide on the future of a player who never quite established himself in north London.