Recently, advancements in large language models (LLMs) have shown an unprecedented ability across various language tasks. This paper investigates the potential application of LLMs to slot filling with noisy ASR transcriptions, via both in-context learning and task-specific fine-tuning. Dedicated prompt designs and fine-tuning approaches are proposed to improve the robustness of LLMs for slot filling with noisy ASR transcriptions. Moreover, a linearised knowledge injection (LKI) scheme is also proposed to integrate dynamic external knowledge into LLMs. Experiments were performed on SLURP to quantify the performance of LLMs, including GPT-3.5-turbo, GPT-4, LLaMA-13B and Vicuna-13B (v1.1 and v1.5) with different ASR error rates. The use of the proposed fine-tuning together with the LKI scheme for LLaMA-13B achieved an 8.3% absolute SLU-F1 improvement compared to the strong Flan-T5-base baseline system on a limited data setup.