The tongue-in-cheek suggestion was made to transport minister Patrick McLoughlin by George Poole, the marketing manager at the Folkstone-based toy factory, who said the company would be prepared to run the troubled line, including the London to Bristol section, free of charge.
In a creative display of his PR acumen, Mr Poole wrote: "I feel there is only one company that stands head and shoulders above the rest.
"This company has been delivering exceptional service on time and to low cost for its customers for years, and should be a serious candidate among the other four bidders."
In a response from the Department of Transport, Mark Reach, private secretary to the minister declined the invitation, but said the wooden trains' "vibrant livery would certainly brighten up the railways".
A spokesman for Bigjigs said the company is considering refining the bid to have another attempt.