Mr Gill has been criticised by some in the party for "double-jobbing".
Asked on BBC Radio's Good Evening Wales programme if he would give up one of the jobs if asked, he replied: "Absolutely, but it has to be for a purpose."
Mr Gill said "it would have to be the leader" asking him to quit a role.
Mr Nuttall won the party's leadership contest on Monday, promising to target Labour voters in the north of England.
He took 9,622 votes with 62.6% of support among party members.
The AM for North Wales sits as an independent in the Senedd after falling out with the UKIP assembly group leader Neil Hamilton.
UKIP's ruling national executive committee had earlier agreed to hold a ballot of Welsh members on whether Mr Gill should continue to be both an AM and an MEP.
But Mr Gill claimed the new leader did not want the ballot to happen.
He told BBC Wales: "I've spoken to Paul about this and Paul tells me he doesn't want a ballot to go ahead. I don't see what it's going to achieve."
Mr Gill said "disgruntled left-behind Labour voters" were a target for UKIP.
He added: "Nigel Farage was able to speak to those people and he is a Dulwich [public school] boy.
"Now if Nigel can speak to those people then without a doubt Paul Nuttall from Bootle on Merseyside absolutely can and will."
Mr Hamilton said: "I think Paul Nuttall has a very good chance of appealing very widely to ex-Labour voters... because he comes from an authentic working-class background in Liverpool.
"I think Paul can win over Welsh Labour voters, perhaps with a bit of help from me and my colleagues in the assembly."
Mr Nuttall previously called for compromise between Mr Hamilton and Mr Gill.
But Mr Hamilton said Mr Gill "doesn't seem to accept that having been elected by the people of north Wales as one of the AMs that he should give his entire focus to this job".
"There is no compromise on that. It's impossible," said the AM for Mid and West Wales.
"It's like pregnancy. You are either pregnant or you're not. You're either full time or you're not."
Mr Hamilton supported Mr Nuttall in the leadership contest. He tweeted he was delighted Mr Nuttall, an MEP for North West England, had won the contest.
Another AM, Mark Reckless, the UKIP member for South Wales East, said Mr Nuttall's comments about targeting Labour voters in the north of England "touched on something with regard to south Wales."
But Labour's Aberavon MP Stephen Kinnock said the British people were "not going to be fooled" by the new UKIP leader.
"Paul Nuttall will use his leadership to fight for a hard Brexit that would put jobs, growth and living standards at risk," he said.
"The best outcome for working people would be to reject UKIP's politics of division, and unite behind a progressive vision for our country that retains our unfettered access to the single market."
Nathan Gill would stand down from one of his two elected positions if newly-elected UKIP leader Paul Nuttall asked him to, the AM and MEP has said.