The Dutchman is chasing an unprecedented 16th win of the season and 51st of his career as well as his fifth in the last six editions of the high-altitude Mexican race.
Verstappen lapped the Autodromo Hermanos Rodriguez with a session best time of 01:19.718 seconds in practice one and slashed it to 1:18.686 in the second hour when there were also some drops of rain.
"I think it was a positive start to the weekend," Verstappen said. "We did look competitive out there so that's good - maybe even better than expected."
Williams' Alex Albon was a surprise second fastest in the first session, 0.095 slower, with McLaren's Lando Norris closest to Verstappen in practice two and 0.119 slower.
"It's close," said Norris, who has been second in three of the last four races and again looked like Red Bull's closest competitor. "A couple of bits still to improve, but we're off to a good start and can build from here."
Home hero Sergio Perez was third and fifth respectively, 0.297 and 0.302 off his team-mate's time but enjoying huge support from the grandstands.
"I would say we didn't have a straightforward day," the Mexican said. "I think overall we are looking in a good position and I think we have a good direction to take."
Ferrari's Charles Leclerc was fifth and third in the sessions with Alfa Romeo's Valtteri Bottas a surprising fourth in the late afternoon.
Mercedes' Lewis Hamilton was only 11th in the first hour but moved up to seventh, with AlphaTauri's Daniel Ricciardo a strong sixth.
Mercedes team boss Toto Wolff explained the slow start to the day by saying the seven-time world champion, disqualified at the previous race, had been testing various items with "lots of measurements on ride height and plank wear."
Hamilton lost second place in Austin for excessive wear of the plank under the car.
"Lewis made a mistake on his fast lap and would have been around about P2 or P3, so that's OK," Wolff said.
Five young drivers replaced race regulars for the opening session as part of a requirement for teams to give track time to rookies, with Britain's Ferrari Academy driver Oliver Bearman the highest placed in 15th for Haas.
Bearman, 18, also set a record previously held by Norris as the youngest British driver to take part in an official Formula One weekend.
Isack Hadjar was 17th for AlphaTauri, Jack Doohan 18th for Alpine, Frederik Vesti 19th for Mercedes and Theo Pourchaire 20th for Alfa Romeo after failing to set a flying lap due to brake issues.
Teams also trialled Pirelli prototype tyres at times during the session, with teams reverting to normal set-up work after two successive sprint weekends with only one practice rather than the usual three.
