The work will help reduce flooding risk to 238 properties in the city centre caused by such abandoned objects, the Environment Agency said.
It coincides with the Destructor Bridge on the river being removed as part of a new development.
On Monday, shopping trolleys were pulled out of the river by the agency.
Matthew Symonds from the trust said: "Navigation can be tricky along the River Avon and no wonder given the amount of stuff that's down there."
The stretch between Windsor and Victoria bridges is being focused on after the agency carried out a survey of flood defences.
An agency spokesman said: "Large objects such as cars and shopping trolleys can cause serious problems when abandoned in a river.
"They can increase flood risk by disrupting flows and cause a hazard to navigation."
The same contractor that is removing the Destructor Bridge will carry out the Â£20,000 river clean up, which is being paid for by the agency.
The Canal and River Trust has appealed for people to stop dumping objects into the river due to the navigational hazards and the impact on wildlife.
Work is expected to be completed by the end of the week.
Cars and mopeds are expected to be lifted from the River Avon in Bath during the first dredging operation for 20 years.