Former Arsenal midfielder Flamini has slipped down the pecking order at AC Milan and was linked with a move to join fellow Frenchman Patrick Vieira at Eastlands in order to boost his World Cup hopes.
Argentina midfielder Gago, 23, is also finding opportunities hard to come by at the Bernabeu, though Real coach Manuel Pellegrini would like to keep him as a squad player.
Real deal: Madrid's Fernando Gago (left) is in demand
Reports in Spain suggest Mancini will have to act quickly if he wants £7.5million Gago as the players agent is said to have agreed a deal in principle for him to return to Boca Juniors on loan.
Vieira has signed until the end of the season but has yet to make his debut as he recovers from a calf injury.
He could be available for next week's Carling Cup semi-final second leg against Manchester United and Mancini told Gazzetta dello Sport: 'Perhaps Real Madrid will give me Gago, maybe Milan (gives me) Flamini.
'I need players. I hope that Vieira is fit for the return.'
Ex-Gunner: Flamini (right) could be back in the Premier League
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