They had plans to offer Carroll for Hatem Ben Arfa but Mike Ashley over-ruled Alan Pardew's wishes of bringing back the Geordie goalscorer by saying: "We have no interest in Carroll".
The Reds have been linked with a move for the France international on a number of occasions in the past and reportedly saw a bid during the January transfer window hastily dismissed by the club.
But L'Equipe report Brendan Rodgers' side have not given up on the former Lyon and Marseille star.
The 26-year-old endured an injury-ridden last campaign, starting just six league games as he struggled with a persistent hamstring problem.
The presence of Joe Kinnear, who was named the club's new director of football in May, could also have an effect on the Frenchman's future, with the former Newcastle manager now running rule on all the club's incoming and outgoing transfers.
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Rio Ave (A) July 20
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MOTHERWELL 2 NEWCASTLE 4 (Vuckic, Gouffran, Sissoko, Marveaux)
Elliot; Tavernier, Williamson (S Taylor, 46), Yanga-Mbiwa (Streete, 82), Dummett; Sissoko (Abeid, 68), Anita (Amalfitano, 82); Obertan (Ferguson, 74), Vuckic (Marveaux, 46), Sammy Ameobi (Richardson, 82), Gouffran (Campbell, 58).
Sub not used: Alnwick.