Honda spent last year with Red Bull's junior squad Toro Rosso, who ended the season ninth in the constructors' championship. Red Bull is hoping that its new partnership will allow it to compete for world championships - something it wasn't able to do in the hybrid era with Renault.
"They have had a strong winter," said Horner. "Red Bull Technology have been working with Honda for 12 months now, having supplied the gearbox and drivetrain solution for Toro Rosso.
"So we have seen behind the scenes their evolution. They are making good progress. Stability of regulations will help them.
"They have settled on a concept and, for the first time since their re-entry to F1, that has continued from one season to the next without it being a completely clean sheet of paper. It will all depend on what the others have done, but certainly versus themselves there has been good progress."
Although Honda made a step pace-wise from 2017 last year, it still lagged behind in terms of reliability. However, Horner says that Red Bull will not be flustered if it is forced to take "a penalty or two" throughout the year.
"Obviously it has been a huge challenge to do 21 races on three engines," he said. "We would rather see consistent evolution, even if that means taking a penalty or two along the way.
"As we saw last year, the impact of those penalties at the right venues can be fairly minimal. So long as the progress and the trajectory is in the right way, there is a lot of optimism within the team, within the factory, and excitement about working in collaboration with the new partner."
Bahrain International Circuit - Winter testing