In the bibliography styles shipped with biblatex, the drivers for the different entry types (@article, @book, ...) typically end with
\newunit\newblock
\usebibmacro{doi+eprint+url}%
\newunit\newblock
\usebibmacro{addendum+pubstate}%
\setunit{\bibpagerefpunct}\newblock
\usebibmacro{pageref}%
\usebibmacro{finentry}}
So one may redefine the doi+eprint+url bibmacro to do nothing, while the finentry bibmacro also undertakes the tasks of the original doi+eprint+url (instead of only typesetting a \finentrypoint).
\documentclass{article}
\usepackage[style=verbose,backref=true]{biblatex}
\renewbibmacro*{doi+eprint+url}{}
\renewbibmacro*{finentry}{%
\iftoggle{bbx:doi}
{\printfield{doi}}
{}%
\newunit\newblock
\iftoggle{bbx:eprint}
{\usebibmacro{eprint}}
{}%
\newunit\newblock
\iftoggle{bbx:url}
{\usebibmacro{url+urldate}}
{}%
\finentry
}
\textheight=120pt% just for the example
\usepackage{filecontents}
\begin{filecontents}{\jobname.bib}
@misc{A01,
author = {Author, A.},
year = {2001},
title = {Alpha},
url = {www.tex.stackechange.com},
}
\end{filecontents}
\addbibresource{\jobname.bib}
\begin{document}
Some text \autocite{A01}.
\printbibliography
\end{document}

Note: The drivers for the @online and @unpublished entry types directly use the url+urdate bibmacro. You can't redefine this macro to do nothing because it is used internally by doi+eprint+url. Instead, you have to copy the definition of @online and @unpublished to your document preamble and remove the respective code lines.
\DeclareBibliographyDriver{online}{%
\usebibmacro{bibindex}%
\usebibmacro{begentry}%
\usebibmacro{author/editor+others/translator+others}%
\setunit{\labelnamepunct}\newblock
\usebibmacro{title}%
\newunit
\printlist{language}%
\newunit\newblock
\usebibmacro{byauthor}%
\newunit\newblock
\usebibmacro{byeditor+others}%
\newunit\newblock
\printfield{version}%
\newunit
\printfield{note}%
\newunit\newblock
\printlist{organization}%
\newunit\newblock
\usebibmacro{date}%
% \newunit\newblock% DELETED
% \iftoggle{bbx:eprint}% DELETED
% {\usebibmacro{eprint}}% DELETED
% {}% DELETED
% \newunit\newblock% DELETED
% \usebibmacro{url+urldate}% DELETED
\newunit\newblock
\usebibmacro{addendum+pubstate}%
\setunit{\bibpagerefpunct}\newblock
\usebibmacro{pageref}%
\usebibmacro{finentry}}
\DeclareBibliographyDriver{unpublished}{%
\usebibmacro{bibindex}%
\usebibmacro{begentry}%
\usebibmacro{author}%
\setunit{\labelnamepunct}\newblock
\usebibmacro{title}%
\newunit
\printlist{language}%
\newunit\newblock
\usebibmacro{byauthor}%
\newunit\newblock
\printfield{howpublished}%
\newunit\newblock
\printfield{note}%
\newunit\newblock
\usebibmacro{location+date}%
% \newunit\newblock% DELETED
% \iftoggle{bbx:url}% DELETED
% {\usebibmacro{url+urldate}}% DELETED
% {}% DELETED
\newunit\newblock
\usebibmacro{addendum+pubstate}%
\setunit{\bibpagerefpunct}\newblock
\usebibmacro{pageref}%
\usebibmacro{finentry}}