September or October had previously been mooted as possible return dates but Maxwell is more optimistic.
"I think it is likely that we will see football in August," he said. "We are seeing it in countries across Europe.
"I think it is definitely achievable, the plans we are putting in place are comprehensive.
"We have to make sure we are giving government comfort that we can do it in a manner that keeps the players, the staff, everybody involved in the game safe, and I'm sure we can do that."
But national clinical director Jason Leitch warned that professional sport would have to go through a number of stages before a return to competitive action.
Leitch said no dates had been attached to the phases.
"They may be three weeks apart if everything goes well, but they may be six weeks apart. It might go backwards a little bit, it might not be everything that happens in phase two that we said," he told BBC Radio Scotland.
"But the professional sports are beginning to get ready for that phase two, which will be the beginning of their major training," he added.
Celtic were crowned champions for a record-equalling ninth consecutive season as the Scottish Premiership campaign was declared over on Monday.
The three leagues below the Premiership were ended last month.
England's Premier League is targeting a restart in June, hoping to follow in the footsteps of Germany's Bundesliga.
Players started training in small groups this week, observing social-distancing measures.
Manchester City defender Benjamin Mendy said he was desperate for the green light.
"I can't wait," said the French World Cup winner. "We are just waiting for the green light. Everywhere we see we are going to start on this date and that date.
"Now I wait for the guys to text us to tell us the league is going to start at this time, and we are going to be so happy.
"But I know they are going to take all the precautions when we are back."