One comes from a political dynasty that has had its differences with President Trump, and the other is intertwined with the president. Now, both are seeking to fill a Republican power vacuum in open-seat congressional races in Texas.

Pierce Bush, the grandson of former President George H.W. Bush, and Ronny L. Jackson, the former White House physician who was Mr. Trump’s thwarted pick to lead the Veterans Affairs Department, announced Monday that they were running for the House of Representatives.

Both entered a crowded G.O.P. field: Mr. Bush in the 22nd District, which covers suburban areas south of Houston like Fort Bend County, and Dr. Jackson in the 13th District, which covers most of Texas’ Panhandle, including Amarillo. The deadline for filing for primary candidates in Texas was Monday.

Mr. Bush and Dr. Jackson are both seeking to appear on the ballot next November with Mr. Trump, whose rift with the Bushes could create an intriguing subplot in 2020 for Mr. Bush, a nonprofit executive.