Use GREP on array to find words

The simplest solution would be to pipe the array elements into grep:

printf -- '%s\n' "${foo[@]}" | grep spi

A couple of notes:

printf is a bash builtin, and you can look it up with man printf. The -- option tells printf that whatever follows is not a command line option. That guards you from having strings in the foo array being interpreted as such.

The notation of "${foo[@]}" expands all the elements of the array as standalone arguments. Overall the words in the array are put into a multi-line string and are piped into grep, which matches every individual line against spi.

The following will print out all words that contain spi:

foo=(spi spid spider spiderman bar)
for i in ${foo[*]}
do
    echo $i | grep "spi"
done

IFS=$'\n' ; echo "${foo[*]}" | grep spi

This produces the output:

spi
spid
spider
spiderman
lospia