Goodwin met with Senate Majority Leader Harry Reid in the Capitol on Thursday.
Sen. Jay Rockefeller has announced plans to retire rather than run again next year. Republican Rep. Shelly Moore Capito has said she plans to seek the seat.
Goodwin served for a few months in the Senate in 2010 after he was appointed to fill a vacancy created by the death of Sen. Robert C. Byrd.
It was not known what other potential recruits Reid has met with, if any.
West Virginia has sent only Democrats to the Senate for decades, but has gone Republican in recent presidential elections.