Connecticut Gov. Ned Lamont will propose spending as much as $5 billion to speed up service on the Metro-North Railroad as part of a larger, 10-year program to overhaul the state’s aging transportation system.

The governor is expected in the coming days to roll out his $18 billion transportation plan, which is expected to include a scaled-down proposal to introduce highway tolls on key crossings and bridges. Mr. Lamont pushed earlier this year for tolls, but failed to persuade lawmakers to bring them up for a vote.

The...