Short Answer: Though it's difficult to say, the NIV is probably correct. It's the narrator, not Jesus, who is most likely speaking in John 3:13-21.
Here's why.
John appears to be establishing a chiasm between the first and second miracles at Cana. This third-person monologue (3:13-21) is paralleled with a similar monologue after the testimony of John the Baptist (3:31-36).
A - First Cana Miracle (2:1-12)
B - Temple Cleansing and events in Jerusalem (2:13-25)
C - Conversation with Jesus (3:1-12)
D - Third Person Monologue (3:13-21)
E - John the Baptist's Testimony (3:22-30)
D' - Third Person Monologue (3:31-36)
C' - Conversation with Jesus (4:1-19)
B' - Replaced Temples and events in Jerusalem (4:20-45)
A' - Second Cana Miracle (4:46-54)
Explanation of the Chiasm
A - A' First and Second Cana Miracles: More than simply linked by location and number, there's a common
subject and sequence that ties these two stories together. (1) A mother
implores Jesus, her son and a father implores Jesus concerning his son
(2:3; 4:47). (2) Jesus rebuffs the request (2:4; 4:48). (3) Both
mother and father continue to press (2:5;4:49). (3) Jesus commands
action without evidence (2:8; 4:50). (4) The servants and the father
obey the command (2:8; 4:50). (5) The miracle is revealed (2:9; 4:51).
(6) The result is belief in Jesus (2:11; 4:53).
B - B' Events at the Temple in Jerusalem: In speaking to the authorities in John 2 Jesus refers not to
the actual temple but rather His own body and to the woman Jesus
declares all such earthly temples now superseded. Both conversations
end with a brief summary of the crowds' response to Jesus' actions. The
later also ends with a reminder of the formers events (4:45).
C - C' Conversations with Jesus: These are the only two sections in John where Jesus has a
conversation with a single individual alone. Both conversations turn
on the issue of water but antithetical parallels also abound: Time of
Day (Night - Noon), Conversation starter (Nicodimus - Jesus), Gender
(man - woman), Ethnicity (Jew - Samaritan), Identity (named -
unnamed)... The list goes on.
D - D' The Third-Person Monologues: Both these monologues begin as narrative speeches in the first
person and jump suddenly into the third person, making it difficult to
tell weather or not the characters are still speaking or if the author
has simply stepped in. In addition, these two discourses share a number
of parallel phrases and related themes.
- "No one has ever gone into heaven except the one who came from heaven-the Son of Man. (3:13)." "The one who comes from above is above all (Jesus); the one who is from the earth belongs to the earth, and speaks as one from the earth (John the Baptist). The one who comes from heaven is above all (3:31-32)."
- "For God so loved the world that he gave his one and only Son, that whoever believes in him shall not perish but have eternal life. For God did not send his Son into the world to condemn the world, but to save the world through him. Whoever believes in him is not condemned, but whoever does not believe stands condemned already because he has not believed in the name of God's one and only Son. (3:16-18)." "Whoever believes in the Son has eternal life, but whoever rejects the Son will not see life, for God's wrath remains on him." (3:36).
E John the Baptist's Testimony: Confined on either side by the third person monologues, John's
testimony appears to represent this chiasm's pivot.
