Citing people familiar with the matter, the British newspaper said the two companies were hammering out the more complex points of the acquisition deal.
It comes after Musk's SpaceX, which operates the Starlink satellite internet service, filed papers for what could be the largest-ever public stock offering.
Starlink says it operates more than 10,000 satellites, offering connectivity to people on the ground.
US tech and e-commerce giant Amazon, founded by billionaire Jeff Bezos, is building a competing network called Amazon Leo, which has more than 200 satellites in orbit.
Globalstar, whose market value is about $9 billion, has its own, smaller "low Earth orbit" satellite constellation.
The newspaper said one factor complicating the lengthy talks was the fact that Apple holds a 20 percent stake in Globalstar, meaning that Amazon and Apple also had to negotiate a deal.
AFP has contacted Amazon, Globalstar and Apple for comment.
The Financial Times report said no deal had been struck and discussions could still collapse.
Globalstar's share price has soared nearly 230 percent over the past year.
In October, Bloomberg reported that the company was exploring a potential sale with several potential buyers including SpaceX.
Globalstar told the Financial Times it "does not comment on industry speculation or rumours," while Amazon declined to respond.
On Wednesday, a source familiar with the matter confirmed SpaceX's confidential IPO filing to AFP, which would reportedly put the rocket and satellite builder on track to list its shares by July.
Media reports have said the SpaceX IPO could be valued at a whopping $75 billion or more. SpaceX did not respond to a request for comment.
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