Parkol Marine is hoping Scarborough Council will allow them to expand their premises to bring more work into the boatyard.
A report written by John Riby, the council's head of engineering and harbour services, said a currently unused joiner's workshop on the Eskside Wharf site is at risk of structural failure.
Parkol Marine, which leases the site from the council, wants to knock the building down to allow them to carry out more repairs and refurbishment.
Other suggested uses for the land include providing public toilets and showers following the demolition of the facilities at the Church Street car park.
In the report it also states the council have been approached by a third party who have declared an interest in using the land.
Parkol owner Jim Morrison said it would be difficult if someone else was to use the land.
He said: "It is a question of space.
"When we have to bring in the crane to move boats it will not be easy if someone is using that property.
"We are the busiest shipyard in the UK and if we had that extra land it would bring in more work.
"We have a tug in the harbour right now that, if the joiners workshop was not there, we could lift out of the water and start work on."
The shipyard is currently working on its largest-ever ship – a 21.5m trawler – but is hoping to start work this week on an even bigger vessel which measures in at 22.8m.
It is estimated that it will cost £14,000 to demolish the old workshop though the leasing of the space could potentially pay for the work.
The report recognises that "the future of the whole Eskside Wharf is inextricably linked to the future of the ship-building company."
As there is interest from other concerned parties the council is bound by its constitution to offer the land for lease, so a tender document will be drawn up to invite offers.
The report will be considered today by the cabinet member for harbour services, Beryl Christon, who will then authorise the drawing up of the tendering document.