"We won't be as ready as we would like but we will do our best," racing director Eric Boullier told reporters after the penultimate day of pre-season testing at the Circuit de Catalunya was marred by further mechanical problems.
"Reliability is still a concern clearly and I don't know how long it is going to take," added the Frenchman, whose British-based team are starting a new partnership with Honda after decades with Mercedes.
"We have some plans in place now, which have been drastically improved in both our organisations working together, and I think it is going to take a few races but we should be at least maybe by Europe ready to be more competitive."
The first race of the European season is Alonso's home Spanish Grand Prix on May 10, the fifth round of the championship.
The season opener is in Melbourne on March 15.
Danish reserve driver Kevin Magnussen was able to complete only 39 laps for McLaren on Saturday before an oil leak halted his progress, with the team deciding to stop and strip the engine.
Button had managed 100 laps on Friday, after just one timed effort on Thursday.
Magnussen, who raced for the team last year and is standing in for Alonso at the test after the Spaniard crashed last Sunday, sounded surprisingly upbeat after his first laps of 2015.
"It is very consistent, very predictable, very smooth in a way and just drivable, whereas last year’s car probably had more downforce, but it was much more sharp and on the limit and unpredictable," he said.
"There is a way to go yet with this car, but it is a completely new design and a new idea and I think it is a very good base. I think it is going to be a very good car, but maybe later in the season."
(Editing by Mark Meadows)
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