Using a reverse proxy to combine two sites by putting one under a URL prefix?

We have an existing web application at www.example.com. The marketing team has a hosted Drupal instance at marketing.example.com. Is there a reverse-proxy solution that would allow me to serve up the marketing content from www.example.com/marketing?

I would like to do this with networking infrastructure changes only; ie NOT changing either the marketing site base url configuration, or adding code to www.example.com.

In Varnish, I can do something like:

if (req.url ~ "^/marketing/") {
  set req.host = "marketing.example.com";
  set req.url = regsub(req.url, "^/marketing/", "/");
} 

However, marketing.example.com is serving up HTML with absolute links like "/blog", that now need to be updated to "/marketing/blog". Could the reverse proxy re-write these links, but keep asking the marketing backend for "/blog"?

Again, I don't want to change the configuration of the marketing site to respond to the new URL format itself. I want the reverse proxy to handle all that.

Right now we're using Varnish, but I'm open to other solutions.