The city of 200,000 people is the capital of Helmand province, parts of which have been seized back by the militant group.
The new assault is their most concerted push yet into the city, in the south of the country.
At least 14 people were killed by a suicide bomber in Lashkar Gah on Monday. including 10 police officers.
The city was the scene of heavy fighting between the Taliban and Nato-led forces before their withdrawal in 2014.
