The Toffees have been chasing the Portuguese ace for some time, but have been kicked into action by rumoured interest from Swansea.
With the South Coast club thought to be considering stepping in with an offer, Martinez is keen to push ahead with his plans to snap up the 23-year-old.
Viana has been a key man in Gil Vicente’s recent impressive run, which has taken them to within touching distance of the European qualification slots in their domestic league.
It is thought that £4.2m will be enough to snap up the skilful wide-man, which has tempted Martinez ahead of the New Year.
The Spaniard is keen to bolster his options in a bid to keep his side in contention for a top six finish, which may be a difficult task due to the quality of the sides in the upper section of the Premier League.
Blackburn were thought to be close to signing Viana over the summer, but elected to pull the plug on the deal.
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