Assuming these are text links and not attachments, first, reset web settings
in IE's tools, options - if that doesn't fix it, go to Help, Detect and
repair in outlook.
If they are attachments, its best to ask the sender to use text links
instead - you can't force outlook to open without saving first since these
are potentially dangerous attachments.

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Diane Poremsky [MVP - Outlook]
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> When I click on a hotlink in a received message (in Outlook 2003) it
> prompts
[quoted text clipped - 4 lines]
>
> I am running Office 2003.
I don't think it prompts you to safe it. Is the dialog called Locate Link
Browser? Then point it to IEXPLORE.EXE

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Robert Sparnaaij [MVP-Outlook]
Coauthor, Configuring Microsoft Outlook 2003
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Outlook FAQ, HowTo, Downloads, Add-Ins and more
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> When I click on a hotlink in a received message (in Outlook 2003) it
> prompts
[quoted text clipped - 4 lines]
>
> I am running Office 2003.