"Without more tests with quick results, it will be difficult to contain this disease and give Americans confidence to go back to work and back to school," Alexander, who chairs the Health, Education, Labor and Pensions Committee, said in a statement.
Alexander's comments come as Trump and several Senate Republicans are signaling they want to quickly start to reopen businesses and other sectors that have been closed by the coronavirus pandemic.
Some Republican lawmakers have warned that the economic damage being sparked by the coronavirus and the subsequent social distancing restrictions outweigh the health impact. The United States has more than 630,000 confirmed cases and more than 27,000 deaths.
"We’ve got to deal with the economic devastation of all of the people who are hurting and I’ll tell you it is time for Texans to go back to work," Sen. Ted Cruz (R-Texas) told a local Texas TV station on Wednesday.