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Are you meaning to send emails from Gmail with an Office 365 email address? If so, please check and confirm the following things are correct:
• The Office 365 email address is a user who has an license enabled Exchange Online mailbox
• The SMTP set to port 587
• Transport Layer Security (TLS) encryption enabled
• The mailbox server name is correct
You can find the server name information by following these steps:
1. Sign in to Outlook Web App.
2. Click Options, and then click See All Options.
3. Click Account, click My Account, and then in the Account Information area, click Settings for POP, IMAP, and SMTP access.
How are things going? Do you have any updates about the issue?
The user has checked the settings and they are all fine. It only happens once every so often...like every 10 emails.
Based on my experience, the issue might be caused by the SMTP settings timeout value, and it should be able to fixed by enlarging the value.