Woodward is the 545th person known to have swum the Strait of Gibraltar, taking four hours and 11 minutes.
The London-based Garden Museum has submitted an application for £3.7m to the Heritage Lottery Fund (HLF) to set up "the country's first archive of garden design".
The £17,000 is part of £120,000 raised to get the project off the ground. The HLF will decide in 2012 and, if successful, the museum will close for development in 2014 and reopen the following year.
The money has been earmarked for storage, a gallery, a study room, digital service, conservation and the employment of an archivist. Woodward said: "This responds to the absence of a collection of modern garden design from the great names of the past decades. People have talked about this for years, but no-one has had a go."
He added that the demand for such an archive became apparent because of interest in Garden Museum exhibitions of designers such as Beth Chatto.
Woodward warned that elderly designers' work could be lost or go abroad. He added that many gardens are lost "within weeks of the creator's death or departure", unlike the work of architects or painters, which is archived at various institutions.
He said he hoped that the Landscape Institute will be involved. It is debating what to do with its collection, which the Garden Museum is storing.
"This is a big challenge for a small museum to take on, but there is an overwhelming and urgent need," Woodward said. He will launch the plan at an event at the museum in late November.