No, it isn't possible in the plain text version to have anything except plain old text characters. That means you can't make links active, or make your text bold or include a logo.
That's because plain text is a format that is the common denominator for even the oldest email clients, and it doesn't have any markup capabilities at all.
However, many modern email clients will detect URLs in plain text and automatically make them links for your subscribers to click on.
If your email is just text with a couple of links and a logo, you could create a very simple HTML version that formats your fonts for you, and lets you add an image and a link or two but still looks like a text document. You'd still need to include a real plain text version without any of that markup though, for people who can't view the HTML.