The influential duo have been sidelined for over two-and-a-half months since picking up injuries in the draw with Newcastle United on September 17.
Bamford - the top English-born goalscorer in the top-flight last season behind Harry Kane - has been out with an ankle problem, with Ayling having been suffering with a knee injury.
The pair have been stepping up their rehabilitation significantly in recent weeks and featured in United's under-23s game at Manchester City on Monday and will be in contention to be involved against the Bees after sitting out the past nine league matches, Bielsa confirmed.
Bielsa said: "It is likely they will be available for the game on Sunday."
The performances of Bamford in scoring 17 times at league level last season and assisting in a further seven goals would greatly increase Leeds's attacking options, with the club having been overly reliant on Raphinha so far this season.
As to whether Bamford, whose form and progress earned him a maiden England call-up in late summer and a debut against Andorra in early September - could be involved from the start on Sunday, Bielsa was non-committal.
Specifically on whether Bamford could start, Bielsa added: "To be available and healthy is the first step for a player who has been in a position out of the side for a few months.
"Nobody just through their presence guarantees a performance and the acquiring of a football fitness level is a step higher to be healthy and training in the sense a player can be healthy and fit. But the adaptation to the competition is something different.
"Sometimes that process accelerates or takes longer, considering the particularities of each player and also sometimes the needs of the team. Sometimes the sporting reality of a team means a player comes back to compete quicker. Managing all of that depends on a lot of factors and I cannot offer a precise conclusion.
"He (Bamford) is a player who scored one goal every two games last season. It is enough to show and signify his importance. Even when he doesn’t score, he is an important part of the way the team functions."
Meanwhile, Bielsa said that Pascal Struijk, who came off at half-time in Tuesday's game with Crystal Palace, should be available.
"We think he will be available for the game on Sunday. He has a knock around his hip. But we think he will be available," the United head coach added.