Amazon.com Inc. executives have made a fresh round of visits to several of the 20 finalists for its $5 billion second-headquarters project, fueling added anticipation as it nears a decision in a process that has stretched over more than a year.

The visits over the past couple of months include New York City, Newark, N.J., and Chicago, according to people familiar with the matter. In addition, Amazon has been following up with other locations, including Miami and the Washington, D.C.-area, according to some of the people. Some...