Hello. I will try to assist you. .
Assuming there is a written rental application, they will typically ask you to disclose employment and income information (all sources of income), credit card and bank account information (creditors, banks, account numbers), and any current creditors. You want to be honest and fully answer. If you are later found to have misrepresented your situation, you can be evicted. Or if a discrepancy is found before a lease is signed, you will be turned down.
Experienced landlords or managers will typically follow up by verifying the information on the application, including background checks. I believe this is what you are concerned about. Soke landlords will call employers to confirm employment and the salary you have stated. Background checks will include rental history, work history, criminal past and a credit check.
Are you concerned about the 401k being discovered or not being discovered? In other words, is this a question of needing your income to be below a certain amount to qualify for the housing or do you need it above a certain amount? If it is the latter, I would volunteer the information in the application, even though the application may not specifically ask for it. If it is something you would rather not disclose, then the only advice I can give is to be honest and direct as to what is specifically asked on the application.