Title contender Hamilton, who is 20 points behind Ferrari's Sebastian Vettel after nine of 20 races, had a similar problem at the previous round in Austria which Bottas won after qualifying on pole.
"His gearbox was damaged in the same way as Lewis's but less severely, and we had hoped to coax it to the end of the cycle," said a Mercedes spokesman. "However, we have not been able to do so.
"The cause of the damage is understood and now behind us, just as for Lewis's problem."
Bottas has taken two poles this season and had been fastest in both Friday's practise sessions, with Hamilton a close second.
The Briton has won for the past three years at Silverstone and is chasing his fifth victory in total at the circuit. Another pole position would leave the triple world champion one short of Michael Schumacher all-time record of 68.
(Reporting by Alan Baldwin, editing by Ed Osmond)
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