Could an author contact you if they received an offer directly from a publisher - and what kind of offer would it have to be to make it worthwhile for an agent?
Sure, I hear from authors with offers in hand all the time.
I've only signed two, and it was a long time ago, when I was eager to build my list.
The problem with offers in hand is that many times the author has already agreed to the terms and there's no way to go back and renegotiate to improve the deal. [Unless you know what standard royalty rates are and the difference between World, World English and what's on the Open Market list for the UK and what's on the Open Market list for the US, chances are you've agreed to something you don't understand.]
The other problem is that with an offer in hand, there's often a deadline for yes/no before an agent can shop the book to anyone else. That is, you can't leave an offer hanging while you go out to try to get a better deal.
And often the deals are with publishers who simply can't sell enough books to have any hope of building the book into good backlist; the publisher doesn't have a distributor, there's no sales or marketing plan in place; there's little or no chance of trade reviews; there's very limited chance of library sales.
Those are all the things I look at if someone arrives with an offer.
And a lot of times, the offer is so low that it just doesn't make financial sense to take it on. Remember, my commission is 15%, so your $2000 advance nets me $300. The opportunity cost for a small deal is what I give up to work on the deal. I'd rather wait for a big deal to pop up than commit myself to a smaller one. In other words, it's not just what I earn, but what I could earn if I took on a different book.
Of course, all this varies by book, author, publisher, and deal. There's no one answer to this question.
Generally a publisher will give you some time before they need an answer because most publishers prefer to have an agent do the deal. There's less to explain, agents understand boilerplate (like all the stuff I mentioned above) and we can handle the author's questions rather than the editor needing to do it.
Of course there's a publisher doing click bait tweets with the quite the opposite point of view
which cracks me up completely.
I had a conversation with a client recently in which the subject was all the ways a publisher could tank a good book and my client was aghast. He had no idea of problems like publishing too quickly, or pricing a book too high just to name the obvious things. The value an agent brings to a book deal for an author is expertise and knowledge. A publisher who doesn't value that, to my mind, is a publisher who either doesn't understand their interests are not the same as the author's, or who do understand and want to exploit an author's lack of knowledge.
But, this is America, and you get to run your business on any moral compass you want to.
And that's actually beside the point of this question so I'll get off my soapbox now, and go back to crushing hopes and dreams (and annoying publishers.)
Bottom line: If you have an offer, email the agent/s you'd like to work with using the subject line OFFER from Publisher/Title. Generally I respond to those emails pretty quickly cause I know you're in a time crunch.