Can I modify outgoing request headers with a Chrome Extension?

Modifying request headers ( https://developer.chrome.com/extensions/webRequest ) is supported in chrome 17.

You are looking at the right place, but intercepting HTTP requests does not exist yet, but the extension team is aware that it's a popular request and would like to get to it sometime in the near future.

Keep in mind that starting from chrome 72, some headers are not allowed unless you add extraHeaders in opt_extraInfoSpec So the above example in @sachinjain024's answer will look something like this:

chrome.webRequest.onBeforeSendHeaders.addListener(
  function(details) {
    for (var i = 0; i < details.requestHeaders.length; ++i) {
      if (details.requestHeaders[i].name === 'User-Agent') {
        details.requestHeaders.splice(i, 1);
        break;
      }
    }
    return { requestHeaders: details.requestHeaders };
  },
  {urls: ['<all_urls>']},
  [ 'blocking', 'requestHeaders', 'extraHeaders']
);

For more info, check the documentation Screenshot from the documentation https://developer.chrome.com/extensions/webRequest#life_cycle_footnote

PS: I am the author of this extension so you can blame me for anything you don't like :)

It was certainly not possible when OP asked the question but soon later Chrome released experimental WebRequest API. But now they have been included officially in Chrome Extension. You can use it modify request and response headers in Chrome.

Look at this example:

chrome.webRequest.onBeforeSendHeaders.addListener(
  function(details) {
    for (var i = 0; i < details.requestHeaders.length; ++i) {
      if (details.requestHeaders[i].name === 'User-Agent') {
        details.requestHeaders.splice(i, 1);
        break;
      }
    }
    return { requestHeaders: details.requestHeaders };
  },
  {urls: ['<all_urls>']},
  ['blocking', 'requestHeaders' /* , 'extraHeaders' */]
  // uncomment 'extraHeaders' above in case of special headers since Chrome 72
  // see https://developer.chrome.com/extensions/webRequest#life_cycle_footnote
);

If you want to use Chrome Extension, you can use Requestly which allows you to modify request and response headers as you wish. Have a look at this snapshot: