The Mail app on your iPhone and others devices iOS can be configured to automatically receive sent Gmail. Messages sent to your Gmail address appear on your iPhone in the Mail app, wherever you are. When you open your mail program, all your Gmail messages are in a separate Inbox. There is no need to wait for the download to finish.
Information in this article applies to iOS 13 through iOS 11, except where noted.
How you set up the Mail app to receive and manage Gmail is slightly different depending on the type your Gmail account — Free Gmail or paid Exchange account.

How to Set Up a Push Gmail Exchange Account in iPhone Mail
Paid Exchange accounts are basically business accounts. To add Gmail as an Exchange push account to iPhone Mail:
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Click » Settings» on the iPhone home screen.
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Select passwords and accounts .
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On the screen » Accounts and passwords» click » Add account» .
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Select Exchange from the options presented.
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Enter your Gmail address in email field . Optionally, add a description to the appropriate field. Then click » Further» .
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In the next window select » To come in» or » Set up manually» . If you choose » To come in» enter your email address to provide your Exchange account information. If you choose » Set up manually» , enter your password and another information manually. Then click » Further» .
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Enter the information requested on the screen to set up an Exchange account, then click » Further» .
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Specify which Exchange folders do you want to send to iPhone Mail and how much messages previous days you want to sync .
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Return to screen » Accounts and passwords» and press » push» Near » Select new data.
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Verify that the Exchange account says » push» or » Fetch» next to her.
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At the bottom of the same screen, click » Automatically» In chapter » Sample to receive the email sent to your Exchange account as quickly as possible. If you would like to receive email messages for a longer period of time, select every 15 minutes , every 30 minutes , or another option.
Set up the free Gmail Push app in the iPhone Mail app
You can also add a free Gmail account to iPhone Mail, where it’s assigned a separate inbox:
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Click » Settings» on the iPhone home screen.
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Select passwords and accounts .
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Click » Add account» on the screen » Passwords and Accounts» .
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Select Google from the options presented.
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Enter your Gmail address (or phone number) in the corresponding field. Click Further .
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Enter your gmail password to the appropriate field. Click Further .
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Specify which Gmail folders you want to send to iPhone Mail.
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Return to screen » Accounts and passwords» and press » push» Near » Select new data.
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Make sure you have a Gmail account next to her written Push or Fetch .
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At the bottom of the same screen, click » Automatically» In chapter » Sample to receive the message sent to your email account as quickly as possible.
Versions of iOS prior to iOS 11 did not have options Automatically . You had to choose from other options, the shortest of which was every 15 minutes .