Mr Weah was among 20 presidential candidates in Tuesday's election.
Sirleaf has been the President of Liberia since 2006 following 14 years of a deadly civil war during the reign of the now incarcerated former President Charles Taylor, who was ousted by rebels in 2003. The former Fifa World Footballer of the Year has long turned a politician with aspiration to be Liberia's President.
In Liberia, a candidate needs more than 50% of the votes for outright victory in an election, with experts forecasting it is likely to go to a second round.
Vice President Joseph Boakai of the ruling Unity Party (UP) is one of the contenders along with former worldwide football star George Weah who is heading up the main opposition Coalition for Democratic Change (CDC). As many as 986 contestants vied for 73 seats in the House of Representatives. Worldwide observers say Tuesday's vote went smoothly despite late starts in some counties.
The election commission is ready to listen to official complaints but vote-counting will continue, spokesman Henry Boyd Flomo said. Final results should be known within two weeks.
Liberia with an estimated population of 4.5 million has 2.18 million registered voters for the elections which were held in 2,080 polling centres nationwide.
The United States called the election "an important step toward achieving Liberia's first peaceful transfer of power from one democratically elected head of state to another in decades". She led the country's recovery from a 14-year civil war and guided this West African country through the Ebola crisis that killed almost 5 000 people.
"All of these people are saying they want change and improvement, and that explains why nearly all of the candidates are presenting themselves as candidates for change", Fomunyoh said.