The couple reportedly then turned their aggression towards crew before staff decided to alert the pilot.
Cockpit crew then radioed ahead to the UK and armed police met the plane at Heathrow's Terminal 5. All passengers except the couple were removed from a rear exit.
A source told The Sun: "The couple were returning from holiday and were tanked up.
"They began screaming at each other and things very quickly got out of hand. It was a blazing row. It was terrifying for all other passengers and crew to endure this obscene behaviour."
Some reports claim the pair had to be restrained, however a spokesperson for BA denied the claim.
The Metro reports that a British Airways statement read: "The welfare of our customers and crew is always our priority. Police were called to meet the aircraft following a dispute between two customers."
Speaking to the MailOnline, a Metropolitan Police spokesman said: "Police were called at 7.43pm to Terminal Five, Heathrow Airport following a report of a disturbance during a flight.
"Officers met the British Airways flight which had come from Los Angeles and landed at 10.10am. Two passengers were spoken to by officers. No arrests were made."