WASHINGTON, May 4 (Reuters) - A fresh push for a bipartisan immigration overhaul, coupled with enhanced border security, is emerging in the U.S. Congress, as thousands of migrants amass across the border in Mexico ahead of the end of COVID-era border restrictions next week.

The latest among those efforts is a last-minute legislative push that would grant U.S. border authorities similar expulsion powers allowed under the expiring COVID restrictions - known as Title 42 - for a period of two years, according to a congressional office involved in the talks.

Separately, the Republican-controlled House of Representatives plans to pass a package of border security measures next week to place tougher constraints on asylum-seekers, resume construction of a wall along the southwest border with Mexico, and expand federal law enforcement.

Many are seeking more sweeping change - but their hopes have been dashed in the past.

It has been 37 years since Congress passed significant immigration reform, but a persistently high volume of migrants and an acute labor shortage have galvanized lawmakers. Republicans also cite the flow of illegal drugs into the United States through ports of entry as reason to harden border security.