Daniel Ricciardo topped the time sheets in the opening running at Hockenheim before team-mate Verstappen led the way later in the day.
Verstappen posted a best effort of one minute and 13:085 seconds to beat Hamilton to the top spot by a little less than three hundredths of a second.
Valtteri Bottas, who earlier on Friday followed Hamilton in extending his stay at Mercedes, finished third.
Championship leader Sebastian Vettel, eight points ahead of Hamilton in the standings, was fourth, a quarter of a second down on Verstappen. His Ferrari team-mate Kimi Raikkonen placed fifth.
Red Bull's pace in practice suggests they should be in a position to challenge Mercedes and Ferrari in Germany this weekend, but their running was not without trouble.
Verstappen lost the best part of 30 minutes with an engine problem, and completed the fewest laps - just 16.
Ricciardo meanwhile, was only 13th in the timings. The Australian is also due to start at the back of the field on Sunday following a series of grid penalties for using too many engine parts.
For Hamilton, he will be bidding to get his title charge back on track after disappointing results in Austria and then at Silverstone a fortnight ago.
Hamilton is also chasing a record-equalling fourth Formula One victory in Germany, bidding to match Michael Schumacher's haul. His rival Vettel has never won at Hockenheim.
While Hamilton would appear to hold an early advantage over his championship challenger, Ferrari typically fail to show their full hand on Friday, and there will be more to come from the Italian team in qualifying on Saturday.
Troubled British outfit McLaren look set for yet another miserable weekend.
Fernando Alonso was only 17th of the 20 runners, 1.7 seconds off the pace, while his team-mate Stoffel Vandoorne was rooted to the foot of the standings in both sessions.
Williams are also in the midst of a woeful year, and Lance Stroll, 18th, and Sergey Sirotkin, 19th, were the slowest on the track apart from Vandoorne.
