“Is it possible to redirect emails from Microsoft Outlook to be received in Gmail or Hotmail?”
You can redirect email from any version of Outlook to Gmail, Hotmail, Yahoo Mail or any other online service. In most cases, you can also respond with the from address you normally use in Outlook, making it appear as though you were responding from your desk.
To make the redirect from Outlook to your favorite email service provider, you need to make a few configuration changes for it to work smoothly.

Configure Outlook to Forward Email
The easiest way to redirect your email from Outlook to Gmail or any other email service provider is to create a forwarding rule in Outlook. From the Outlook menu, go to Tools > Rules and Alerts. Click on the New Rule button.
The rule you create is going to be Apply this rule after the message arrives through the specified account. Click on the specified link and choose your email account.

Click next and choose forward it to people or distribution list. Here again you need to click on the people or distribution list link and enter your Gmail or Hotmail address.

You can add exceptions to your rule, like say if you never want email from your boss forwarded to your Gmail account, or if you’re on a discussion list with dozens of messages a day that you only want to see when Outlook is open.
Another useful option is to check the box to mark forwarded messages as read so you don’t see them as new messages next time you open Outlook.
Once you have all the customization for your Outlook Rule, click the Finish button. This rule will be active as long as Outlook is running on your computer. If you get your Outlook mail from an Exchange Server, this rule should work even if Outlook is closed on your computer.
Change Reply Address in Gmail
In Gmail, you can make forwarding your mail more useful, you can send mail from Gmail as your Outlook email address. Go to Settings, click on the Accounts link and Add another email address you own. If you include your Outlook email address here, you can then send as your Outlook email address for all the mail that forwards from Outlook.

Don’t forget you can also sync Outlook Contacts and Calendar to Gmail.