The Big Ten ultimately sourced the tests from a private company instead, the officials said.
The conference announced Wednesday its football season -- on hold due to the outbreak -- would resume on Oct. 23. It said it would utilize "stringent medical protocols," including daily testing of its student-athletes and coaches.
Trump had since last month been publicly insisting the Big Ten kick off its football season, and he spoke with the conference's commissioner, Kevin Warren, on Sept. 1, about the matter.
"I called the commissioner a couple of weeks ago, and we started putting a lot of pressure on, frankly, because there was no reason for it not to come back," Trump told reporters Wednesday.
After that call, Trump directed White House staff to provide any federal resources the conference needed, according to the senior administration official.
"Probably for political reasons, it was easy for the Big Ten to convince their presidents to vote for it, if it wasn't going to be provided by this White House," the official said.
Another senior Trump administration official on Thursday said the White House did offer “all our resources” to the Big Ten, “including test availability.” The official said the testing offer did not delve into specifics, such as the particular type of tests the White House would potentially supply.
Trump has for months called for the return of professional and college sports, many of which had been put on hold. He has pushed for states and schools to lift coronavirus-related restrictions despite the continued high rate of virus transmission in certain parts of the country; his own presidential campaign has ignored local restrictions on crowd sizes, mask-wearing and social distancing.
Many leagues have recently resumed play with safety protocols restrictions and limits on spectators. They have had varying degrees of success in responding to athletes who have fallen ill with the virus.
The Big Ten said Wednesday that athletes, coaches, trainers and others who go on the field would get tested daily and that athletes who receive a positive result would receive a second test to confirm the result.
If both tests were positive, the athlete would have to undergo cardiac testing and receive clearance from a cardiologist before they could return to competition -- at the earliest 21 days after his initial diagnosis.
Trump's interest in the Big Ten was spurred by calls from players and their parents for the season to resume, the senior administration official said.
The official said the White House had made hundreds of calls on the topic with Big Ten coaches, officials, athletic directors, parents and players.