Everton are hoping to extend Landon Donovan's loan spell at Goodison Park until the end of the season.
The United States captain has made a positive impact since joining the Merseyside club on a short-term deal in January.
Donovan is due to return to parent club Los Angeles Galaxy, where he recently signed a new contract, to prepare for the new Major League Soccer season later this month.
Moyes has previously stated that any extension to the 10-week loan would be unlikely, but he has now made contact with Galaxy to see if it is possible.
Moyes said: "I am keen to keep Landon and it sounds initially in America that they might say yes. I think Landon is keen to stay also so it looks as though all parties are keen.
"But I have not got that confirmed yet, I can't say exactly if that is going to happen. If we can make it happen we will.''
Donovan's move to Everton, which has coincided with an upturn in the team's form, has benefited both club and player.
Moyes added: "In the main he has done a really good job for us. It is very rare you can bring in someone in January on loan who can make a real difference to you but Landon has. He came in when the team was on the up, so that has suited him, but we cannot deny he has done well.
"Landon had to be thrown straight in so we had to find out very quickly if he could do a job or not. But I think his manager at LA Galaxy would say the best World Cup preparation for him would be to start in their team.
"They will be keen to get him back so we will have to be respectful of their wishes. If we can keep him it would be nice but if not I totally understand it.''
If no new agreement is forthcoming, Sunday's clash with struggling Hull will be Donovan's last at Goodison Park.