diff --git a/base/mpfr.jl b/base/mpfr.jl
index c85531856f5c9..cd8c8092e0720 100644
--- a/base/mpfr.jl
+++ b/base/mpfr.jl
@@ -294,7 +294,14 @@ function round(::Type{T}, x::BigFloat, r::Union{RoundingMode, MPFRRoundingMode})
     end
     return unsafe_trunc(T, res)
 end
-round(::Type{BigInt}, x::BigFloat, r::Union{RoundingMode, MPFRRoundingMode}) = _unchecked_cast(BigInt, x, r)
+
+function round(::Type{BigInt}, x::BigFloat, r::Union{RoundingMode, MPFRRoundingMode})
+    clear_flags()
+    res = _unchecked_cast(BigInt, x, r)
+    had_range_exception() && throw(InexactError(:round, BigInt, x))
+    return res
+end
+
 round(::Type{T}, x::BigFloat, r::RoundingMode) where T<:Union{Signed, Unsigned} =
     invoke(round, Tuple{Type{<:Union{Signed, Unsigned}}, BigFloat, Union{RoundingMode, MPFRRoundingMode}}, T, x, r)
 round(::Type{BigInt}, x::BigFloat, r::RoundingMode) =
diff --git a/test/mpfr.jl b/test/mpfr.jl
index 86c7d345f49fd..54ec018327459 100644
--- a/test/mpfr.jl
+++ b/test/mpfr.jl
@@ -667,6 +667,10 @@ end
     @test typeof(round(Int64, x)) == Int64 && round(Int64, x) == 42
     @test typeof(round(Int, x)) == Int && round(Int, x) == 42
     @test typeof(round(UInt, x)) == UInt && round(UInt, x) == 0x2a
+
+    # Issue #44662
+    @test_throws InexactError round(Integer, big(Inf))
+    @test_throws InexactError round(Integer, big(NaN))
 end
 @testset "string representation" begin
     str = "1.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000012"