Linking Advice Needed

Hi Guys,
I hope some of you may be able to help me here!
I have built a site which is SEO optimized for certain keywords.

I have a few keywords that I would like the site to rank for but although the site is relevant to the keywords it is not optimized for the phrases.

My question is, if I was to build links to the site like this:-
<a href="http://www.mysite.com">targeted keyword</a>

Is it possible that with enough links that the site will rank for the targeted keyword even though the pages it links to are not optimized for that. The site as a whole is relevant to the keywords though.

Hope you can help

Yes, it's certainly possible. It will take a lot of links though.

My site ranks #35 for a keyword with 8m results and doesnt include the keyword in the title, and includes it only twice on the whole site. One of my subpages ranks #8 for a keyword with 1.7m results, which doesn't include the keyword in the URL, and only includes it three times on the page. I concentrate soley on off-page SEO.

Thanks, I just wanted to check before I spent time building links etc.

Is there a reason you can't add pages to your site optimized for the other keyword? If they don't quite fit with the other keywords or topics on the site, you could use a silo structure to isolate them on your site.