The London Legacy Development Corporation met on Tuesday to discuss West Ham's bid to occupy the stadium, which was given "preferred bidder" status last December. Since then the two sides have been trying to resolve details of the club's proposed move, including how much they will pay towards conversion costs for the £429 million stadium.
Sources say talks between the two sides in the private meeting were positive, and London's Mayor echoed those sentiments afterwards. "There has been a lot of progress, I won't hide that," he said.
"Good conversations have been going on. Tax payers put a huge amount of money, £500,000, in to building an iconic stadium, and they put further money in to doing it up.
"We have got to get the best possible terms for the tax payer."
Sources said the club hoped a deal could be signed within weeks rather than months. That will come as comforting news to organisers of the 2015 Rugby World Cup, who need to know by the end of next month whether they will be able to use the stadium.
"It would be great to do it," Johnson added when asked about World Cup matches being staged at Stratford.
"We are looking at it. I have asked our engineers to see whether we can make it work, but this particular negotiation isn't relevant to that."
The LLDC will discuss the future of the stadium at their next board meeting on March 26, although they can call an extraordinary general meeting if they reach an agreement with West Ham before then.
Should a deal be agreed, West Ham could move in to the stadium at the start of the 2016-17 season.