I think you meant it the other way around, like this:
string input = "This is a string and may or may not contain a word we are looking for like Pete";
List<string> search = new List<string>() { "Pete", "Lynne" };
bool found = search.Any(s => input.Contains(s));
The last line is equivalent to:
bool found = search.Where(s => input.Contains(s)).Any();