Trump had made immigration a cornerstone of his presidency and promised, during his election campaign, that he would build a wall along the US-Mexico border.
The former Republican president said the wall was needed to curb illegal immigration and drug trafficking across the southern border.
Democrats, however, called the wall immoral, ineffective and expensive.
Biden, a Democrat, has so far opposed completing the wall. The construction would prove extremely difficult as much of the border terrain is rugged, hilly terrain while other areas are controlled by private land owners.
However, Department of Homeland Security Secretary Alejandro Mayorkas told his employees that he may give the go-ahead for work on the wall to plug "gaps" in the current wall, The Washington Times reported citing notes it reviewed from a recent discussion between Mayorkas and employees of Immigration and Customs Enforcement.
“It’s not a single answer to a single question. There are different projects that the chief of the Border Patrol has presented and the acting commissioner of CBP presented to me,” Mayorkas reportedly said.
“The president has communicated quite clearly his decision that the emergency that triggered the devotion of DOD funds to the construction of the border wall is ended. But that leaves room to make decisions as the administration, as part of the administration, in particular areas of the wall that need renovation, particular projects that need to be finished."
Filling "gaps," installing "gates" and adding technology to parts of the wall, which have been finished but not outfitted, are among the work that needs to be done.
Biden has not always opposed building boundaries at the border. When he served as a senator, he voted to approve 700 miles of border fencing. Also, during his tenure as vice president under former president Barack Obama, 100 miles of that fencing was under construction.
In spite of his pledge not to build "one more foot" of the wall, he might have no other choice but to complete its construction given the fact that Congress has already allocated money to the project.
Based on the Impoundment Control Act, when Congress allocates money to a project, it must be completed, unless questions of efficiency arise or a president requests that the project is revoked.
So far, Biden has not submitted a revocation request.
Each year over the past four years, Congress has allocated $1.375bn to the project, which is still far from complete.
Critics, however, fault the new administration for unravelling strict border protections put in place by Trump, including measures that kept migrants in Mexico while they waited for their cases to be processed.