The United States captain has made a huge 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.
Captain fantastic: Barring the odd big miss, Donovan has excelled at Goodison
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.
The 27-year-old has sharpened his game in the Barclays Premier League in a World Cup year while Moyes has found him a good quality short-term solution to some of his squad's injury problems.
The midfielder has added an extra dimension on the right side of Everton's attack and Moyes views the move as a complete success.
Fan base: A facebook group has been created for fans to show their appreciation
The only low point was a bad miss from in front of goal in last week's defeat at Tottenham.
Moyes said: '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.'32
If no new agreement is forthcoming, Sunday's clash with struggling Hull will be Donovan's last at Goodison Park.
On the mend: Tim Cahill could return to action
Moyes is hoping he can be joined in the team by midfield talisman Tim Cahill, who is battling to overcome a calf injury.
The Australian was initially ruled out for up to three weeks after limping out of the first leg of the Europa League tie against Sporting Lisbon 17 days ago.
'He is getting closer,' Moyes said. 'He has not trained yet but there is a chance he might train tomorrow. If he does that would give him a chance.'
Top scorer Louis Saha, however, is still missing with the hamstring injury that kept him out of the trip to Tottenham. Phil Jagielka, recently returned from a long-term knee injury, could be doubtful with a thigh strain, as could Leon Osman with a foot problem.
Moyes added: 'We hope to have roughly the same players we had last week. In the main I hope the majority of them will make it.'
All of Everton's international players returned from midweek duty unscathed and Moyes was particularly pleased to see left-back Leighton Baines make his England debut on Wednesday.
The Scot said: 'He has deserved it, he has played consistently well for Everton. It took him a little while to get started in the side but he was patient and got through, and is now showing his worth.'
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