Gareth Bale was an unused substitute in Real Madrid's Champions League final win over Liverpool last month
Forward Bale, 32, is currently a free agent having left Real Madrid after nine years in Spain.
LAFC co-president John Thorrington said negotiations with Bale had not been "a financial conversation" and expects the deal to be finalised "within 10 days".
"We don't have a firm date," Thorrington confirmed.
"Our transfer window opens in early July when we can register [players]. We still have to go through paperwork, hopefully a week or the next 10 days to take care of that, to welcome him and his family to LA, get up to speed."
Thorrington would not be drawn into the details of the agreement in place with Bale, but said that it would run beyond November's World Cup in Qatar, where Wales will be making their first appearance at a finals since 1958.
"I'd love to get into specifics but I don't as a policy, but what I can tell you is that LAFC and Gareth are hoping that this is a long-term partnership," Thorrington said, speaking after a 2-0 win over New York Red Bulls that kept the side top of the MLS standings.
"It was never about six months just for the World Cup. We were always talking about longer term than that. That was very clear from Gareth and his representative.
"When Gareth is looking at what he needs to do, his priority will be at LAFC - we are absolutely sure of that - but we're not naive to the fact that he, like some other players of ours, have a World Cup on the horizon."
Thorrington, who is also Los Angeles FC's general manager, said that Bale's first game for the MLS side could potentially be the home derby against LA Galaxy on 9 July.
The five-time Champions League winner had been linked with a return to Tottenham Hotspur as well as a move to home-city club Cardiff.
The signing of Bale, who joined Real Madrid for a then world-record fee of £85m in 2013, follows LAFC's move for Italy defender Giorgio Chiellini earlier this month.
"We presented to them what we feel is an unique opportunity for them at a stage of their career to do something different," Thorrington said.
"But we want to win and we want winners, and we have a guy who's won five Champions Leagues and a guy who's won the World Cup [Chiellini].
"We have two great players who will continue to do great things in the next phase of their career here.
"We see them being great influences on the guys we already have here."
There was speculation that Bale would end his playing career had Wales missed out on qualification for the 2022 World Cup.
But after Robert Page's side booked their place in Qatar by beating Ukraine in their World Cup play-off final earlier this month, the only question about Bale's future was which club he would join.
There had been suggestions Bale would opt to join Cardiff, with his agents confirming last week that they had held talks with the Championship club.
Cardiff share a training ground with the Football Association of Wales, which would have allowed Bale to continue working closely with Wales' medical and fitness staff while on club duty.
But he is instead heading for the MLS, where the season runs from February until October and the competition's secondary transfer window opens on 7 July.
"We have done incredible diligence and it was an open, transparent conversation and we quickly saw it was a right match in the face of stiff competition worldwide - and he [Bale] agreed and we're excited to welcome him in the next couple of weeks," Thorrington said.
"It wasn't a financial conversation with Gareth or us. If Gareth was making this decision with financial motives he wouldn't be in MLS.
"This league is really exciting globally and he saw an opportunity to make a real impact. He sees here an opportunity to make a difference with the stage we provide.
"He sees the chance to pull this sport up to competing with the other sports.
"As much as possible... understanding how to manage a player like Gareth, a Ferrari or thoroughbred racehorse, they need extra attention in order to succeed, we're confident we'll be able to do that.
"The benefit of him joining the group we have, we will have a player who can play as much as he can, but we will do so sensibly.
"His priority will be at LAFC but we're not naive - he has a World Cup on the horizon and how we manage his preparation ahead of the World Cup will be important to Gareth and the success of LA, but we want him peaking after the game with the US!
"The exact preparation he needs will be the same for us as it is for the World Cup, which I know for Wales is an historic one."
Wales will head to Qatar in November, where they will face the USA, Iran and England in Group B at the World Cup.
LAFC, who are managed by former USA international Steve Cherundolo, are currently top of the MLS' Western Conference and have 18 regular-season games to play.
When asked what Bale and Chiellini will bring to his squad, Cherundolo said: "Having been on phone calls and zooms with Gareth Bale and Chiellini, I know 100% they are here for the right reasons.
"I hope he [Bale] will bring what he brought to Tottenham, Real Madrid - goals, assists, constant threat - a prolific player who knows how to win championships."
The end-of-season play-offs follow, with the season-ending MLS Cup final to be played on 5 November - 16 days before Wales' World Cup campaign begins with a game against the USA.
-- Courtesy of BBC Sport