When New Jersey Gov. Chris Christie canceled a Hudson River tunnel project in 2010, it wasn’t just billions of dollars in potential funding that was lost.

Also swept away were years of planning by a tangle of local, regional and federal bureaucracies run by politicians with competing agendas.

Five years later, political leaders and transportation officials are talking about starting another tunnel project amid questions over whether they can finish the job to avert a potential transit nightmare.

Mr. Christie, a Republican seeking his party’s nomination for president, is due to sit down Tuesday with U.S. Transportation Secretary Anthony Foxx to discuss an Amtrak plan that includes two new Hudson River tunnels, estimated to cost $15 billion to $20 billion.

Mr. Christie’s meeting with Mr. Foxx follows finger-pointing after electrical problems in Amtrak’s pair of existing tunnels last month led to several days of delays for NJ Transit commuters.