There are various synonyms for the phrase "paying court to", which essentially means to show intention or interest in establishing a romantic relationship. Some of these alternatives include the terms "wooing" and "courting", which imply the act of seeking someone's favor or affection by flattering them or offering attentions. Meanwhile, the phrase "making advances" can denote an act of creating opportunities for a possible romantic connection. Other possible synonyms include "pursuing", "chasing", "seducing", and "romancing". These words generally carry connotations of actively seeking to win someone's affection or partnership. However, it should be noted that the tone and context in which they are used can affect the meaning and intention behind them.