Offering the asking price would certainly imply your interest and keenness to anyone. I find that some people offer a bit less, looking for a bargain. Offering asking price means you're a straight player... everything down the line. Offering more... to me, as a Landlord... is something I would never entertain... it doesn't tempt me... I've had it happen, I've never gone that route. However, things that have worked as tie-breakers for me (apart from my gut feel about someone) is whether they seem capable in life and are ready to go.By this I mean... be clear on when you can move in. Make it known that you won't mess around with returning referencing information. Possibly (depending on what the Deposit level requested is) offer more Deposit (up to the legal maximum). Let it be known that you already know what is in the area, for example, or that you know the Council Tax band... let it be known you've done some research. However, all that said... this would affect me as a Landlord who does their own viewings... who can say with an Agent?Now... you're only viewing it tomorrow... you don't yet know for sure it's perfect for you in every way... so many really aren't. So please go in with a critical eye and assess sensibly afterwards. Never get caught-up in a property you won't even own. Heck, never get emotionally invested in a property you want to own... but much less so one you're going to rent.The crux of my answer is - "I wouldn't" - certainly not tomorrow. If a bidding war ensues then you could consider it (I wouldn't because I wouldn't do it as a Landlord, but you could). Hope that helps.
I have never been offered more money,and I am not sure how I would react.It might make me wonder if the person had been turned down before,and was a bit desperate.Someone that I would always favour is a person looking to settle in,as opposed to renting there while they decide to buy somewhere or get a better job etc.Longterm tenants are highly sought-after.