Some 2011 models did not ship from the factory with Internet recovery in their ROM. You can only use Internet Recovery when the firmware was updated to include that feature. If that was never done, you are stuck without it unless you can install a version of macOS that fixes that.
Internet Recovery is NOT available in ROM in most Macs before 2011 models. So you will need to explore this list of other possibilities.
When your computer was released, the way you launched the required Utilities (including Disk Utility and Installer) was to use the ones on the Release software DVD. if you have a model-specific version for your model (unlikely) or a Full Retail 10.6 DVD —and a graphic card supported by early 10.6 — you use its Utilities, boot and install that version, then use Software update to get to 10.6.8 with all updates, which is the first version that can reach out to Apple servers and download and install a later version.
If you have replaced the graphics card, so that it is one that does NOT have Drivers included in the release DVD, you can boot the release DVD, but you get no picture. if you have the old card lying around, you could re-install the old card to use the Installer DVD.
10.11 El Capitan is a recommended waypoint, even if you expect to install a later version. 10.11 allows encrypted Internet connections and Mac App Store access. That makes getting later versions much easier.
The next source of Utilities is the Recovery Partition on the boot drive. If your drive spins up, even if not MacOS bootable, it may still have a usable recovery partition. To get there, try invoking recovery with Command-R or … hold Alt/Option at startup and see if the recovery partition shows as a potentially bootable drive.
Recovery Partitions up through 10.12 Sierra can be found with the Startup Manager (Alt/Option boot). At 10.13 High Sierra, if an SSD boot drive is used, the format is transitioned to APFS. The Recovery partition is present, but it is inside the APFS container, and the Startup Manager on an older Mac may not be able to find it.
The next source of Utilities to consider is any MacOS 10.6 or later versions on any additional drives or clones you may (or may not) have lying about, even if they are from another Mac. You can use those Utilities to ERASE a new drive, and start the installer to place MacOS on the new drive.
The next source to consider is a Time Machine backup drive. Versions from 10.7.3 or later are said to contain a Recovery Partition that could be used to ERASE a new drive and run Installer to place MacOS on a new drive. Time Machine backups created in MacOS 11 Big Sur or later are APFS format, and APFS format backup drives do NOT have a Recovery partition.
Two Mac solutions:
With certain combinations of new and old Mac, you can use Target Disk mode to repair, erase, and install on the drive of the old Mac, by treating it as a disk drive on the new Mac.
Transfer files between two Mac computers using target disk mode - Apple Support
IF you have a different Mac, you can use it to download MacOS install image, then interrupt the process and create a BOOTABLE USB-stick Installer/Utilities stick. BOOTABLE is key, because the way you will install from this USB-Stick is to BOOT the USB-stick, and use its Utilities to ERASE your drive and start the Installer. here is the article on bootable USB-Stick Utilities/Installer:
What you need to create a bootable installer
• A USB flash drive or other secondary volume formatted as GUID partition Map, Mac OS Extended, with at least 14GB of available storage
• A downloaded installer for macOS Big Sur, Catalina, Mojave, High Sierra, or El Capitan.
The Terminal command assumes that Installer in located in the /Applications folder.
from:
How to create a bootable installer for macOS
Create a bootable installer for macOS - Apple Support
NB>> if you name your incoming USB stick exactly MyVolume, you can copy and paste the very long Terminal command from the article directly into the Terminal window, without having to change anything.
