Should we raise our offer on a house we like, or wait for the seller to “see the light”?

We’re currently negotiating on a house we actually really like, and I’d love some second opinions.

The house is listed at £590k and has been on the market for just under three weeks. We viewed it quickly and made an offer of £533k about 2 weeks into the listing. We’re chain-free FTB in England, mortgage AIP, big deposit ready, solid buyers.

The seller’s agent initially said the seller might go down to £580k, but didn’t counter-offer formally. On a call, my husband and the agent both agreed that we might just sit tight and see if the seller “sees the light” i.e. realises he may not get anything better.

But I’m starting to feel like this is risky. Should we be the ones to raise our offer to, say, £540k? It would be the top end of our next move, and we’re willing to go to £550k max, but no higher.

My fear is that if we increase our offer now, we’re just bidding against ourselves and shown our hand too early. But on the other hand, I worry that waiting could backfire and someone else might swoop in. I want to do a firm good faith negotiation without playing games.

Would you raise now to £640k, or hold your ground and wait for the seller to come back? How long for?