MOUNTAIN VIEW, Calif. — Apple is moving into Silicon Valley headquarters that look like a spaceship. Facebook is expanding its campus with a new building designed by Frank Gehry. Now it’s Google’s turn.

This week, Google, the search giant, is expected to propose new headquarters — a series of canopylike buildings from Heatherwick Studio, a London design firm known for works like the fiery caldron at the 2012 Olympics, and Bjarke Ingels, a Danish architect known for his innovative designs.

The project in Mountain View, which Google has not made public but has discussed with members of the City Council, is likely to aggravate an increasingly testy relationship between the company and community leaders who fear the company is overrunning their small city.

When Google moved here in 1999, it had a dozen employees and a search engine known only to computer aficionados. Now, its 20,000 local employees make it the biggest employer in a city that is bursting at the seams.