Hi,

To accomplish this you will have to create a Custom Profile for the users that you want to give the ability to Transfer Records. This will allow them to transfer any records that they have Read Access to.

Give permission to Transfer Records

In Professional, Enterprise, Performance, Unlimited, and Developer editions, System Administrators can manage and create profiles for their organization. Profiles make it easier for Administrators to manage the security and access settings for a group of users by creating a single profile and assigning multiple users to that profile

Salesforce Classic:

To create or edit a user profile:
Navigate to Setup | Administration Setup | Manage Users | Profiles
Click the New button (custom profiles are based on existing profiles)
 Click Save and set the appropriate permissions and Page Layout for the profile.
To set the permissions for the new Profile:

Navigate to Setup | Administration Setup | Manage Users | Profiles
Click on Edit next to the new custom Profile you created
Check the box next to 'Transfer Records'
Click Save
To set the permissions for the new Profile using the Enhanced Profile editor: 

Navigate to Setup | Administration Setup | Manage Users | Profiles
Click on the name of the new custom Profile you created
Click on System Permissions
Check the box next to 'Transfer Records'
Click Save
Lightning Experience:

To create or edit a user profile:

1. Click on the Gear Icon | Select Setup | Type in "Profiles" on the Quick Find box
2. Click the 'New Profile' button (custom profiles are based on existing profiles)
3. Click Save and set the appropriate permissions and Page Layout for the profile.  

To set the permissions for the new Profile:

1. Click on the Gear Icon | Select Setup | Type in "Profiles" on the Quick Find box
2. Click on Edit next to the new custom Profile you created
3. Check the box next to 'Transfer Record'
4. Click Save  

To set the permissions for the new Profile using the Enhanced Profile editor: 

1. Click on the Gear Icon | Select Setup | Type in "Profiles" on the Quick Find box
2. Click on the name of the new custom Profile you created
3. Click on System Permissions
4. Check the box next to 'Transfer Record'
5. Click Save


For your reference, you can check the below article too,

https://help.salesforce.com/articleView?id=account_owner_transfer.htm&type=5


I hope you find the above information is helpful. If it does, please mark as Best Answer to help others too.

Thanks.