Short answer: Create a drip campaign that triggers when a contact custom field is mapped to the deal stage changes to the appropriate deal stage.
If your contacts can have more than one deal open at a time, this is the article for you:
Create a single value picklist custom field on a contact that will hold a deal stage value.
Administration->Account Settings->Custom Fields [Add Custom Field Type]
Be sure to list the deal names exactly as you see them in Pipedrive. If you have more than one pipeline, you may have to add the pipeline name before the deal stage, one per line.
If you have not done so, set up a crm update job
Contacts->CRM Contact Updates [New]
In your update job, map deal stage name to your custom field.
Create a new drip campaign (Automation->Campaigns->Drip Campaign
Select the Contact Changed starting step
Set your query for the starting step