The Baggies made a move for Hulse earlier this month, but their initial bid was rejected.
The player himself rejected a move to Cardiff City - but Millwall are also in for him and have had a bid accepted, while Norwich have also made a bid in writing.
West Brom are now favourites to land Hulse, who they want to spearhead their promotion push.
The Hawthorns outfit confirmed they had made a new bid saying: "Albion have signalled their intent to land Rob Hulse by tabling an improved bid for the Crewe hotshot.
"The Baggies' second, undisclosed offer was lodged this afternoon (Tuesday) and the club are now awaiting the Railwaymen's response."
