WASHINGTON—President Trump is likely to sign the border-security legislation finalized Wednesday that would keep the government from closing this weekend, people familiar with his plans said, but he continues to look for ways to supplement the modest funding it contains for barriers.

Mr. Trump said he would take a serious look at the legislation, which includes far less money for barriers on the U.S.-Mexico border than he has demanded, but gave no assurance he would sign it. He said he doesn’t want to see another government...