Is it permissible to continue speaking with a university colleague, while suspecting mutual feelings exist, for the purpose of encouraging him to propose, especially since he is shy?

It is obligatory not to be complacent in speaking with a non-mahram young man, because it is a gateway to temptation and a means of corruption. The claim that a girl speaks with her colleague considering him like her brother is a false claim. Many scholars have gone to the extent of prohibiting conversation between a young woman and a non-mahram man, even if it is merely exchanging greetings, if there is a fear of temptation. It is permissible for a woman to offer herself to someone she sees as righteous, to marry her, within certain controls and etiquette. If he proceeds to marry her, then this is good; otherwise, she must disengage from attachment to him. As for a girl striving for a relationship with a young man for him to marry her, this is contrary to Islamic law. The lawful method is for the young man to propose to her from her family, and then marry her through a valid marriage contract. It is not permissible to communicate in seclusion before the marriage contract, nor to exchange love messages, due to the temptation involved in that.