Donald Trump’s legal team has rejected the special counsel Robert Mueller’s latest terms for an interview with the president as part of the Russia investigation, and put forward a counter-offer, according to US media reports.

“We have now given him an answer. Obviously, he should take a few days to consider it, but we should get this resolved,” Trump’s lawyer, the former New York mayor Rudy Giuliani, said on Wednesday on a radio show hosted by a fellow Trump lawyer, Jay Sekulow.

Mueller and his team have been seeking to grill Trump as they investigate Russian interference in the 2016 election, as part of the effort to determine whether anyone on the Trump campaign colluded with the Russian efforts and whether Trump has obstructed justice.

Trump has said he wants to sit down with the investigators but his lawyers have resisted, fearing he could end up accused of lying to investigators based on his statements in an interview.

Giuliani said negotiations over a potential sitdown should wrap up by the end of the month.

“We do not want to run into the November elections. So back up from that, this should be over by September 1,” he said in the radio appearance. “If there is going to be an interview, let’s have it. If there’s not going to be an interview, let him write his report.”

The special counsel’s team and the Trump lawyers have long wrangled over the terms of a potential interview.

In their latest letter to Mueller’s investigators, the Trump team rejected Mueller’s proposed interview terms but suggested narrow conditions under which Trump would agree to talk, the New York Times reported. The lawyers did not want Trump answering questions about potential obstruction of justice, the report said.

“He has got all the information he needs. The interview would provide nothing in addition to what he already has so he can write his report. And we have been willing to cross it, in other words we have been willing to say this is the answer he will give,” Giuliani said. “And we’ll be stuck with it.”

Trump has a history of making false statements in public, and his lawyers worry he could get himself in trouble by answering questions in the Russia inquiry. Giuliani has warned the questions could be “perjury traps”.

Trump, meanwhile, has railed against the investigation as a “witch-hunt” and demanded that it come to an end.

If the sides don’t reach an agreement for a voluntary interview, Mueller could subpoena Trump to force him to answer questions – a move that would probably set off an unprecedented legal battle.

Giuliani said in May that “we don’t have to” answer a subpoena – arguing that Trump could assert the privilege of his office to quash such a move.

Legal experts said Giuliani’s argument was unlikely to succeed, but the ultimate decision could be made by the supreme court.

Bill Clinton was subpoenaed to testify during the investigation into his relationship with Monica Lewinsky, but he ultimately agreed to testify voluntarily so the order was not put to the legal test.

The high court ruled that executive privilege could not be used to avoid a subpoena for White House tapes in a 1974 case involving President Richard Nixon.