I think that Linus is too heavy for a very cheap "CPU".
If by Linux you mean "something easy to develop on", then most micro-controllers come nowadays with an Arduino compatible layer.
Arduino-compatible layer is useful when you don't want to setup real IDE or your own build system for native toolchain. For ease of development it does not buy you that much over straight C for anything remotely non-trivial. On the other hand many small MCUs can run RTOSes with somewhat POSIX-ish APIs (particularly notable in this space is ESPxx, for which the official toolchain is port of FreeRTOS and probably nobody uses it as bare hardware)
If you remove the Linux part from your wishlist, you can consider some "NodeMCU" boards on Chinese websites, eg: https://www.banggood.com/Wemos-NODEMCU-ESP8266-12F-WiFi-1_30... https://www.banggood.com/Wemos-TTGO-WiFi-Bluetooth-Battery-E...