I know a way how to add " Print on Check" as a field on a specific report in QuickBooks Desktop, Ebenidze.

The "Print name on check as" field is unavailable on any report. We can follow the recommended solution of 

BigRedConsulting. Just create a custom field in the vendor profile in the additional info tab. From there, enter the Print on Check for your vendor. Here's how: 

1. Go to the Vendor menus, then select the Vendor Center.
2. Double-click any profile on the list to open and edit it.
3. Select the Additional Info tab. Then select Define Fields.
4. Name your custom field in the Label column.
5. In the Use for the column, select the checkbox to add the custom name field to vendor profiles.

Once done, you can run a Vendor Contact List report. Then, add the custom field in the report. Let me show you how:

1. Go to Reports on the top menu.
2. Choose Vendors & Payables.
3. Click Vendor Contact List.
4. Tap the Customize Report tab.
5. Hit the Display tab.
6. In the Columns field, type Print on Check. 
7. Press OK. 

In this particular situation, I recommend sending a message to the product engineers to let them know you'd like to have the option to add " Print on Check" as a field to any report. Here's how to get your feedback submitted:

1. Go to the Help menu and select Send Feedback Online.
2. Click Product Suggestion.
3. Choose a feedback type and product area on the QuickBooks Feedback window.
4. Enter your suggestion or feedback, name, and email address (optional).
5. Select the Send Feedback Online.

For future reference, read through this article: Customize reports in QuickBooks Desktop. It helps you learn about how to use filters so you can get the details you want in a report.

Let me know in the comment section if you need more help in accomplishing your other QuickBooks tasks. I'll be right here to keep assisting. You have a good one.