Suppose an AdWords advertiser runs ads that target one of my sites and they get good, consistent results. They like the content, and the established visitor base, and think that it would be a great place to place paid ads. If they approach me directly and offer to pay for regular ads on the site is it a problem with the AdSense or AdWords TOS?
If I understand correctly, you mean that the advertiser would know that his ads are being displayed on your page? 1) How would they know this? 2) I don't believe it is against AdSense TOS although do not quote me on it.
Sorry, I wasn't more clear. In this scenario the advertiser would pay me directly for a totally separate ad featuring his website/product placed somewhere else on the webpage. The AdSense ad blocks would continue to run as usual, though he probably would stop running the AdWords ad targeted at my website.
I don't see any problems with it. The ads will be paid or will be added to your site like any other banner advertisement or paid link, or external link on a page. It shouldn't have any problem with adsense.