Dyke also suggested Spurs could play their Champions League home matches at the national stadium next term, with their qualification for the competition already assured.
Tottenham are building a £400million, 61,000-capacity ground adjacent to their White Hart Lane home. They plan to move into the new stadium at the start of the 2018/19 campaign.
According to Dyke, Wembley is on the verge of being confirmed as Spurs' base for the season before the move.
Next term's home Champions League fixtures could also be played there given White Hart Lane's capacity is likely to be reduced from its current 36,284 to facilitate construction of the new ground.
"We are certainly in discussion with Spurs that they should come in for the full season when they're rebuilding their stadium," Dyke told Sky Sports HQ.
"I think we're a long way down the path on reaching an agreement.
"I think there's some discussion about whether they'll play their Champions League games there at Wembley next year but I don't know much about that.
"On the full season, I think we're quite close to a deal."
Dyke also said Chelsea could play home games at Wembley in the near future.
The Blues have plans to redevelop their Stamford Bridge ground, which could leave them needing a temporary home for several seasons.
"I think Chelsea might come for a longer period," he added.
"Chelsea have got really exciting plans for their stadium. They're going to demolish the whole thing and build a whole new stadium on the existing site.
"If they came (to Wembley), that would mean them coming for three years."
Tottenham confirmed they are looking at where to play European games next season.
The club did not mention Wembley, but in a statement said: "In order to comply with all UEFA stadium requirements for staging Champions League fixtures, we are discussing options that include the possibility of playing our European games away from White Hart Lane next season."
Around 4,000 seats will be lost at White Hart Lane next season, Tottenham clarified.
The club said in the statement: "The impact of any works that might be undertaken in the North East corner would result in a reduction in seats for games next season of approximately 4,000 in total when general admission and corporate seats are included."
The Tottenham Hotspur Supporters' Trust welcomed the news from Dyke and expressed hope that the short-term Wembley deal would soon be finalised.
"Spurs fans' preferred option for the club's year away from White Hart Lane has overwhelmingly been Wembley," the trust said in a statement.
"That's been backed up by two surveys the Supporters' Trust conducted showing over 80% support for that option. And THST has campaigned publicly for Wembley for some time.
"So it's good to hear Greg Dyke say discussions are 'a long way down the path'. At the moment, nothing seems to be confirmed, but we hope fans will get some facts soon."