A common limit is within one month
The rent start date is when rent begins. It may also be called the contract start date or move-in date.
Even when negotiation is possible, staying within about one month after applying is the usual range. Some management companies require the date to be within two to three weeks.
Why it is hard to delay
For the owner, holding a vacant room for too long increases vacancy loss. If another applicant can move in earlier, that applicant may be preferred.
This is especially true for popular or ready-to-move-in properties.
How to negotiate
Give a specific desired date and reason, such as your current move-out date or job start date.
Also prepare for some overlap in rent, because the requested date may not be accepted.
Summary
A one-month delay is usually the upper range, and two to three weeks is common with many managers.
Confirm the possible rent start date before applying.