The Portugal international has urged his club to move for the Argentine, with Leonardo Jarim hinting that he could try to sign the Benfica star in the close-season
Liverpool and Manchester United are among the clubs thought to be interested in the 26-year-old Argentine, who has spent the last four years in Portugal with Benfica.
Moutinho himself swapped the Primeira Liga for the principality outfit last summer and has urged Gaitan to follow suit this time around.
"He would be a great signing. He's an excellent player," he told reporters.
"I know him well and if he comes here, his quality would help us. If not, we still have a good group."
Monaco coach Leonardo Jardim, meanwhile, has hinted that he could move for Gaitain, saying he is keen to add an attacking midfielder to his squad.
"We are seeking an attacking midfielder and Gaitain meets those characteristics. His name is on the table," he said.