The open letter comes as Moderate leader Ulf Kristersson is potentially next in line to receive the mandate to explore forming a centre-right government, following the failure of Social Democrat leader Magdalena Andersson.
Sweden can't afford political gridlock that is "allowed to go on indefinitely", Åkesson said in the letter.
"It is against this backdrop that I am now reaching out directly to you. With this letter, I would like to invite you to a meeting and a no-strings-attached conversation on the parliamentary situation and the deadlocks that clearly exist. Not least between our respective parties," he wrote.
The Centre Party is strictly against collaborating with any government that includes the far-right Sweden Democrats. Åkesson stressed that the invitation to meet does not mean that his party is dropping its demand for cabinet seats.
"Elisabeth, we do not need to agree on Sweden's political direction to talk to one another about Sweden's parliamentary situation. On the contrary, it is perhaps precisely when differences are at their greatest that the need to meet is most obvious," he wrote.
Speaker of parliament Andreas Norlén is set to hold a press conference on Monday afternoon to outline the next steps forward in the process of forming a government.
The Centre Party has so far declined to comment on the invitation, but Karin Källström, a lawyer described in Swedish media as an "independent Centre Party opinion writer" argued that the party should consider joining a government dependent on SD support.
"The Centre Party should be be pragmatic and think 'is it possible to do a deal with the left-wing parties and if so would that be good for Sweden," she told the Aftonbladet newspaper. "And what is the alternative with Ulf Kristersson. I personally believe that you could get through a lot more policy by negotiating with the Tidö parties, even if this was with the support of the Sweden Democrats."
In an interview on Swedish television on Sunday night, Justice Minister Gunnar Strömmer, from the Moderate Party, said he did not believe the invitation would unlock the possibility of a government on the right.
"I do not perceive that there has been any dramatic shift that changes the fundamental starting point," he said. "There is a lot that unites the parties on the right, and if you take that as the starting point, then of course there are possibilities, but then, as everyone known, there are these deadlocks."
"Exactly how these can be broken, and if it's even possible to do so is a question that mainly concerns the parties themselves."