The Arsenal legend took charge of Genoa earlier this season after the dismissal of Alberto Gilardino. Vieira has Genoa sitting in 12th place on the Serie A table, 14 points clear of the bottom three.
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Sky Italia says Roma has Vieira on its radar.
However, the capital club are also watching five more names: Stefano Pioli, Maurizio Sarri, Vincenzo Montella, Roberto Mancini and Massimiliano Allegri.
Veteran Claudio Ranieri was appointed coach to the end of the season.
Vieira has a contract with Genoa that runs until the summer of 2026.