Nearly all the results from Tuesday’s poll have been counted, the election commission says.
Mr Weah, the first African to win the Ballon D’Or football award, is leading with 39%, while Mr Boakai is in second place with 29%.
A second round between the pair is expected next month.
They lead the field of 20 candidates who competed to succeed Ellen Johnson Sirleaf, Africa’s first female elected president and a winner of the Nobel Peace Prize.
Fewer than 5% of polling stations have yet to declare results, and lawyer Charles Brumskine is in third place with 9.8%.
Both Mr Weah and Mr Boakai had predicted they would win the first round of voting.
Mr Weah’s former manager on the football field, Arsene Wenger, was earlier this week apparently duped by false reports that he had already been elected president.
He told reporters: “It’s not often you have a former player who becomes president of a country. So well done, Georgie.”
Meet the front runners
George Weah, 51: Former Fifa World Footballer of the Year
Arsene Wenger, now at Arsenal, was his coach at Monaco in the 1990s
Has the political backing of jailed warlord and former president Charles Taylor
Taylor’s ex-wife, Jewel Howard Taylor, is his running mate
Joseph Boakai, 73: Nicknamed “Sleepy Joe”
Denies it is because he is often caught napping at public events, says it is because he is a dreamer
Vice-president under Ellen Johnson Sirleaf since 2005
Has distanced himself from her record, saying “a lot more needs to be achieved”
Liberia, which was founded by freed US slaves in the 19th Century, has not had a smooth transfer of power in 73 years.
Ms Sirleaf took office in 2006, after her predecessor, Charles Taylor, was forced out of office by rebels in 2003, ending a long civil war.
Taylor is currently serving a 50-year prison sentence in the UK for war crimes related to the conflict in neighbouring Sierra Leone.
Mr Weah, 51, has chosen Taylor’s ex-wife Jewel Howard Taylor as his running mate.