The 21-year-old made 41 appearances last season and grabbed the goal that sent the Glovers to the second tier for the first time in their history.
And with a number of established internationals blocking Burn's path to Martin Jol's starting XI Johnson is looking at borrowing the defender for another season.
"Quite rightly they (Fulham) are saying they need everyone to come back for pre-season to work with them," Johnson said.
"We'd like to get things done early but we are holding to Fulham as they pull all the strings. What I do know is that Danny Burn would be very comfortable with coming back to us, that is the main thing.
"We are sure that his agents and his representatives do as well so with a bit of luck and a bit of negotiation hopefully we will get him back."
Burn was the only loanee in the starting line-up that clinched promotion against Brentford at Wembley and Johnson is focusing on adding new signings to his squad before turning his attention to the loan market.
"You try to get in the permanent ones first," he added. "Any loan I get will be a long-term loan anyway."
Meanwhile Johnson confirmed that Seth Nana Ofori-Twumasi, a player he managed at Northampton and Peterborough, will join the Glovers in pre-season in a bid to prove his fitness.