Managing personal Outlook.com account settings, security, and privacy
Hello,
You can certainly add a gmail address as an "alias" for your Microsoft account. However, an alias is just a symbolic identifier/username. It doesn't mean Microsoft account can now receive emails on Gmail's behalf. It doesn't work that way.
To add the alias, sign into your personal account via this link: https://account.live.com?
Go to Your Info > Sign in preference, Add an alias.
Add existing: this will allow you to add a 3rd party email address, such as yahoo, gmail, as long as they are not already associated with another Microsoft account.
Once added, make this new alias your primary alias if you want it to be primary.
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Outlook.com is the home mail box for alias using Microsoft's native domains only, such as outlook.com, hotmail.com, msn.com, live.com. It is not an email client app that can host other email service provider. So it cannot be used to check Gmail.