The 36-year-old double Formula One world champion is out of contract at the end of the season and has said that he needs to know the former champions have a competitive car before committing.
The Spaniard won his 2005 and 2006 titles with Renault, who announced on Friday a three-year deal to supply McLaren with engines from 2018, and remains one of the sport's top drivers despite the last three years in an uncompetitive car.
"We’re going to turn our attention to Fernando now," Brown told reporters after the engine announcement at the Singapore Grand Prix. "We’ve been speaking a lot.
"He’s won his two championships with Renault, he’s very happy with the racing team, we’re very happy with him.
"So, I think, in the not too distant future we should be able to get something done and that’ll be the last piece of the puzzle."
McLaren's other driver, Belgian rookie Stoffel Vandoorne, has already been confirmed but Alonso is central to their planning.
Renault's engine is still behind Mercedes and Ferrari in performance, but ahead of Honda, and has powered Red Bull's Australian Daniel Ricciardo to victory this season.
Brown, a former racer and marketing expert who replaced McLaren’s veteran boss Ron Dennis at the helm late last year, will also be busy hunting for sponsorship.
Honda contributed over $100 million a year to McLaren’s budget, according to insiders, and their departure leaves a significant shortfall.
McLaren, the sport's second most successful team in terms of race wins, have not had a title sponsor since telecoms company Vodafone left at the end of 2013. They last won a grand prix at the end of 2012.
Brown was confident the gap could be bridged and said the team -- largely owned by Bahrain's Mumtalakat Holding Company and Saudi-born businessman Mansour Ojjeh -- was prepared to bear the short-term financial pain.
"We’re fortunate in that we have very committed shareholders, that the remit is to go win races, make the right sporting decisions and then the business will follow," said the American.
"It’ll take some time but our shareholders are very committed, very aware and we wouldn’t have made this decision if we weren’t prepared to work through short-term economic loss from the lack of Honda income."
(Reporting by Abhishek Takle, editing by Alan Baldwin and Toby Davis)
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