Brad Gilbert says he will wait until the end of Wimbledon before deciding whether to take up the chance to coach Britain's Andy Murray.
Andy Murray has the ability to go a long way in the game
Gilbert, the former coach of Andre Agassi and Andy Roddick, is currently working as a commentator for ESPN.
"At the end of the tournament, we'll see what happens but nothing is imminent," Gilbert told BBC Sport.
LTA chief Roger Draper said his organisation would fund the move, but Murray said it was not "a done deal".
"Nothing's been set yet," Murray told BBC Sport.
"He's one of a few coaches I've been speaking to but it's not a done deal."
The 44-year-old Gilbert was hailed as "the greatest coach of all time" by Agassi, a player he helped win six Grand Slam titles and an Olympic gold medal.
He also guided Roddick win the US Open before parting company with him in late 2004.
"I get mentioned in connection with a lot of jobs. I'm flattered but, for the moment, I'm working for ESPN," added Gilbert.
"A lot will depend on my family. I have a wife and three kids and my son's going to go to college. These are all things I need to think about and figure out - I can't tell you this moment.
Andy Murray split from his last coach, Mark Petchey, in April
"I've had a chance to play, I've had a chance to coach and I've had a chance to do TV - I'm almost 45 years old and I've never worked a day in my life.
"I get to get up and do tennis, that's my passion and I do enjoy coaching a lot.
"I haven't watched Andy play a lot but from what I've seen he's got a lot of talent and he has the ability to go a long way in the game."
Gilbert enjoyed a successful playing career, reaching fourth in the world rankings in 1990.
But it is as a coach that Gilbert has really excelled, making him one of the most in-demand in the world - and Draper said he accepted Gilbert's services would not come cheap.
A reported salary of £500,000 would test the LTA's resolve, but Draper said he would be prepared to pay for an acknowledged coaching star.
"If you're going to get the best coaches in the world, you're going to have to pay for them," Draper, who said he would also like Gilbert to work with other young British players and help develop coaches, told BBC Sport.
"We would be willing to pay whatever it would cost to get Andy into the top 10 and drive British tennis and improve confidence and belief across the sport.
"If we can get people like Brad on board that will be fantastic, not just for Andy, but for British tennis.
"We've been looking at the resources in the LTA, the board have been working hard at reallocating those resources because we actually want to spend more on the players and the coaches.
"I think he does want Brad as his coach but there's a lot of detail to go through in terms of how that might work in the future."
Andy's a certain character so you have to get to know him and what makes him tick
Murray's former coach Leon Smith
Leon Smith, the man who coached Murray for six years until he was 17 years old, told BBC Sport that his former charge needed a big-name coach.
Smith teamed up again with Murray at this month's French Open on a temporary basis.
"It will make a big difference to him having a top, top level coach because I think he'll listen to them," he said.
"I think he's waiting for someone to really talk with him and shape his game to the next stage."
Murray was 11 years old when he started working with Smith, and last year he teamed up with Mark Petchey before the pair parted company in April.
"I think he wanted someone in place for Wimbledon but that wasn't to be," said Smith, who is now an LTA national coach.
"Where he is right now, with all the expectations on him, I think it would be good to have a big name coach that could shoulder some of that responsibility and can deal with the press.
"It's a big responsibility and also he's a certain character as well, so you have to get to know him and what makes him tick.
"That's where it's important to have continuity - to build the relationship."