I haven't exactly followed the Hawkeye solo series on a monthly basis but I have recently purchased and read the first volume of the Hawkeye paperback. So, if there are any developments past issue #5 I wouldn't know anything about it.
As I was reading the book I came upon this page:
Clint asks Kate for assistance but she keeps on asking, "Why me?" In return, Clint replies with different answers but none pleased her.
Being as experienced with women as Clint is, he probably figured out what Kate wanted to hear and quickly stopped her questions by cutting her off and saying, "Because I don't want to sleep with you?" As shown in the seventh panel of the image above, Kate immediately got mad as if disappointed with his response then just as fast deflected it with humor. Afterwards after a little heart-to-heart Kate agrees to help Clint but at the end of the page this is shown:
It could be argued that the captions that appeared in these last two panels are Clint's thoughts and not Kate's own. Though, because the captions are placed on a panel with only Kate in it means that whatever is written in the caption also applies to her whether it's from Clint or not.
In my experience with other comics, other mediums and real life, Kate's upset and disappointed look on her face indicate that she is greatly saddened when Clint said, "Because I don't want to sleep with you?" Adding "Dummy" right after is a way to insult someone that hurt your feelings (just to ease the pain) something that most, if not all, do.
Kate isn't so subtle in hiding her attraction, however, as shown in this picture:
Which registered as "Bad idea four" but, he is most likely referring to Kate driving. Or is he?
The way Kate always hangs out with Clint is also peaks my interest. Joining him in social occasions such as his rooftop barbecues with the tenants of his building.
She has other friends and, not to mention, extremely rich, why does she choose to hang out with him?
A little later Clint gets kidnapped by SHIELD right from where he is standing. Which resulted into Kate waiting in his apartment until three in the morning, as shown below:
And who can forget about this little escapade in Madripoor?
Nothing truly awkward is happening here until I found out that Madame Masque is really ...
... Kate Bishop in disguise.
While reading this, the main thing that came to my mind is that the real Madame Masque would probably get one of her henchmen to get the credit card for her and relieve herself of the shame of doing that particular dirty work.
This image below also hints on a possible romance:
Hand-on-hand, an effective way of saying admiration and, possibly, love through a picture. Especially if the other is in bed sick and/or injured.
Some might say that Kate Bishop is way too young/underage for Clint Barton. I say, by the time of the first issue of the new Young Avengers series, Kate is already eighteen-years-old (or seventeen turning eighteen soon) because the theme of the current is no longer about a group of sixteen-year-olds wanting to be an adult but eighteen-year-olds experiencing the first stage of being an adult (in the case of Noh-Varr early twenties?). So, she is too young but, not young enough that it would be fiendish or illegal.