Representative Elijah Cummings has reportedly been trying to meet with Trump for a month to discuss the issue, "and it became the subject of a testy exchange between Trump and reporters last month".
After a public exchange of words about the meeting, there was doubt about whether it would occur.
Trump speculated at the time that someone like Senate Minority Leader Charles E. Schumer - or "some other lightweight" - had told Cummings not to have the meeting.
"I said when you talk about the African American community, you make it sound like it's the most very bad place to be", Cummings told reporters after the meeting. Cummings agreed but was surprised when Trump later held a surprise news conference and accused Cummings of backing out of the meeting.
Rep. Jason Chaffetz, a Utah Republican and chairman of the committee, sent the letter along with committee member Rep. Elijah Cummings, a Maryland Democrat.
House lawmakers expressed concern over the White House's digital record keeping practices on Wednesday.
And, Welch said, "the price is starting to kill us, we just can't afford it". "I was actually looking forward to meeting with the president about the skyrocketing price of prescription drugs".
He also told CNN he looked forward to working out a time to meet with the President and he wasn't anxious about the Trump's comment, waving it off as bad information that someone on his staff gave him.
"This meeting was first talked about when I ran into the president at his inaugural luncheon", Cummings said.
Trump tweeted the day before his meeting with Cummings and Welch that he is "working on a new system where there will be competition" in the drug industry. "You go to a drug store in January, it's $100".
"What we have is a lot of pharmaceutical companies that are raising the prices of drugs unreasonably".
Cummings, the ranking Democrat on the House Oversight and Government Reform Committee, dismissed Trump's allegations that he was wire-tapped by former President Barack Obama - telling Blitzer that the dust-up may cause him to be more careful about his Twitter use.
Welch noted that the president has publicly supported efforts to lower drug prices.