In general, it is difficult for conventional adaptive interference cancellation schemes to improve speech intelligibility in the presence of interference whose source is obtained asynchronously with the corrupted target speech. This is because there are inevitable timing drifts between the two inputs to the system. To address this problem, a drift-compensated adaptive filtering (DCAF) scheme is proposed in this paper. It extends the conventional schemes by adopting a timing drift identification and compensation algorithm which, together with an advanced adaptive filtering algorithm, makes it possible to reduce the interference even if the magnitude of the timing drift rate is as big as one or two percent. This range is large enough to cover timing accuracy variations of most audio recording and playing devices nowadays.