Infantino is under pressure from three confederations to resign, including Europe's UEFA, Asia's AFC and CONCACAF, which represents North and Central America and the Caribbean, after his aborted proposal to sell a stake in the World Cup.
Infantino, who could potentially face a vote of no confidence from world soccer's regional powerhouses, is visiting the Dominican Republic this weekend and was reportedly planning to attend a competition organised by the Caribbean Football Union (CFU).
In a letter sent to Infantino on Friday and seen by Reuters, Montagliani cited the "governance crisis surrounding FIFA" as reason for the FIFA president to reconsider attending the CFU U-14 Series event.
"As you may be aware, your proposed attendance has garnered significant media attention," Montagliani, who is also a vice-president on the FIFA Council, wrote.
"...the media will focus on this situation, which will unfortunately significantly take away from the technical nature of the CFU youth competition.
"As such, I respectfully ask you in the name of football to reconsider your attendance as it will unfortunately undermine the technical nature of this important youth development activity."
Reuters has contacted FIFA for comment.
Infantino was received at Punta Cana by the Dominican Republic's Minister of Sports and Recreation Kelvin Cruz.
"We deeply appreciate your visit to the Dominican Republic, without doubt that your presence in our country puts us on the international football radar and opens us great prospects for growth and development of Dominican football," Cruz said in a statement.
Infantino was also welcomed by Dominican Football Federation president Jose Deschamps, National Commissioner of Football Benny Metz and Olympic Committee President Garibaldy Bautista Lorenzo.
Infantino is seeking a fourth term until 2031 but several national associations and key officials have withdrawn support for the Swiss administrator.