"Let no foreigner who is bound to the LORD say, “The LORD will surely exclude me from his people.”... And foreigners who bind themselves to the LORD to minister to him, to love the name of the LORD, and to be his servants... these I will bring to my holy mountain and give them joy in my house of prayer... for my house will be called a house of prayer for all nations.” (Isaiah 56:3-7)
From a Christian perspective, Jesus is the Chosen One and those who acknowledge him, whether Jewish or Gentile are God's "chosen people". Jesus means "anointed" or "Chosen one".
"Here is my servant, whom I uphold, my chosen one in whom I delight; I will put my Spirit on him and he will bring justice to the nations." (Isaiah 42:1)The phrase "chosen people" is used exclusively of the followers of Jesus in the Christian scriptures.
The Apostle Peter quotes Moses, insisting membership of God's people is defined on the basis of faith not race:
"For Moses said, ‘The Lord your God will raise up for you a prophet like me from among your own people; you must listen to everything he tells you. Anyone who does not listen to him will be completely cut off from their people." (Acts 3:22-23; Deuteronomy 18:15, 18, 19).The Apostle Paul similarly insists:
"A person is not a Jew who is one only outwardly, nor is circumcision merely outward and physical. No, a person is a Jew who is one inwardly; and circumcision is circumcision of the heart, by the Spirit, not by the written code." (Romans 2:28-29)To believe this is patently not anti-Semitic any more than to suggest those who deny Jesus is their Messiah are anti-British. Anti-Semitism is racism and must be repudiated just as much as Islamophobia.
To read what the New Testament actually teaches about God's "Chosen people" see Israel and the Church: Who are God's Chosen People?
For a simple introduction see: Seven Biblical Answers to Popular Zionist Assumptions