Sign changes of Kloosterman sums and exceptional characters

We prove that the existence of exceptional real zeroes of Dirichlet $L$-functions would lead to cancellations in the sum ${\sum }_{p\le x}\text{Kl}(1,p)$ of Kloosterman sums over primes, and also to sign changes of $\text{Kl}(1,n)$, where $n$ runs over integers with exactly two prime factors. Our arguments involve a variant of Bombieri’s sieve, bounds for twisted sums of Kloosterman sums, and work of Fouvry and Michel on sums of $\mid \text{Kl}(1,n)\mid$.