The timeline leaves the door open for Howard to return in time for the team's Sept. 10 regular-season opener against the Baltimore Ravens.
Earlier Monday, coach DeMeco Ryans had categorized Howard's injury as something that would keep the 27-year-old "out for a while."
Howard recently signed a three-year, $56 million extension that included $36.5 million guaranteed before the start of training camp, but he injured his hand at practice Saturday.
In 2022, Howard started 17 games and was seventh among offensive tackles in ESPN's pass block win rate (93.1%).
Veteran swing tackle George Fant, who was signed by the team less than two weeks ago, now will get the opportunity to start at right tackle while Howard recovers.
Fant was brought to Houston as insurance for the offensive line after playing for the New York Jets from 2020 to 2022 in the same scheme Texans offensive coordinator Bobby Slowik runs.
In 2021, Fant had a pass block win rate of 89.3% through 15 games, 31st among tackles, before suffering a knee injury that landed him on injured reserve.
Howard's injury is the second for the Texans' offensive line in the past few days. Center Scott Quessenberry was placed on injured reserve after he tore an ACL and MCL during Thursday's practice.