John Hadland, 34, was on a break with his family when he disappeared at around 10pm on Wednesday in the resort of Puerto del Carmen, on the island's south-east coast.
He is originally from Holt Park and has recently been living in Armley.
His brother Jamie said he became involved in an argument with his relatives before leaving their hotel alone without his wallet, passport or mobile phone.
They were due to leave the island on a flight in the early hours of Thursday morning but Mr Hadland did not show up at the airport and has not been seen or heard from since.
His family have now returned to the UK and reported his disappearance to the Foreign and Commonwealth Office. Local police have also been informed.
"He is on his own in Lanzarote and has no phone, money or passport. He has been reported missing but still no news. Support would be much appreciated," said Jamie Hadland.