The invitation comes less than five months ahead of his visit to Cuba as part of his tour around North America.
The meeting between the two leaders will take place "in a strictly private manner" in the study of the Paul VI audience hall, Fr Federico Lombardi, Vatican spokesperson said, according to The Tablet.
In a statement, Fr Lombardi said President Castro "publicly thanked the Pope for his role in fostering the rapprochement" between Cuba and the US.
In December last year Barack Obama and President Castro said their nations were working towards re-kindling diplomatic relations, and both men credited Pope Francis for his help in securing a deal.
The Holy Father used letters and private meetings at the Vatican between US and Cuban representatives last year to begin to re-establish relations.
The Pope is expected to travel to Cuba before his three-city tour of the United States, which is due to take place at the end of September.
The official itinerary has yet to be released, however this will be the second visit of a pope to Cuba in three years; Pope Benedict XVI visited in 2012.
Visiting both Cuba and the US on the same trip is seen to represent the Pope's desire for smoother relations between the two countries.
The US first imposed a trade embargo on Cuba in 1960, as a response to the Marxist revolution that put Fidel Castro in power.
