Or use a dplyr & stringr solution. But in principle its the same idea:
library(tidyverse)
library(stringr)
df %>%
mutate(result=str_detect(df$string,gsub(" ", "|", df$substring)))
substring string result
1 my new phone this is a mobile phone TRUE
2 She would buy new phones Yes, I have two phones TRUE
3 telephonessss my old telephone FALSE
4 telephone234 telephone234 TRUE