Venn Diagrams and Patterns

Something like this (it's just a rough proposition):

\documentclass[border=5pt,tikz]{standalone}
   \usetikzlibrary{patterns}
%\usepackage{venndiagram}
%
\begin{document}
%\begin{venndiagram2sets}[shade=orange,tikzoptions={red}]
%   \fillA
%\end{venndiagram2sets}
\begin{tikzpicture}[red,every node/.style={font=\Large}]
    \draw[pattern=north west lines] (0,0) circle(1.5);
    \draw (2,0) circle(1.5);
        \node[fill=white,inner sep=1pt,rounded corners,below=3] at (90:1.5) {$A$};
        \node[xshift=2cm,below=3] at (90:1.5) {$B$};
\end{tikzpicture}
\end{document}

The output:

EDIT: Here is a proposal with horizontal lines (manually pattern drawn – just for fun):

\documentclass[border=5pt,tikz]{standalone}
   \usetikzlibrary{patterns}
%\usepackage{venndiagram}
%
\begin{document}
%\begin{venndiagram2sets}[shade=orange,tikzoptions={red}]
%   \fillA
%\end{venndiagram2sets}
\begin{tikzpicture}[red,every node/.style={font=\Large}]
\begin{scope}
    \clip (0,0) circle(1.5);
    \foreach \x in {-5,-4.95,...,6}
    {
        \draw[black,yshift=\x cm] (-5,\x) -- (5,\x);
    }
\end{scope}
    \draw (2,0) circle(1.5);
    \draw (0,0) circle(1.5);
        \node[fill=white,inner sep=1pt,rounded corners,below=3] at (90:1.5) {$A$};
        \node[xshift=2cm,below=3] at (90:1.5) {$B$};
\end{tikzpicture}
\end{document}

Here is the output:

I agree with marmot's comment that the best option is to make the package more flexible (my suggestion would be to replace fill=\@venn@shade with something like every venn region/.try). In the meantime, here's a fix that uses a scope to add extra options to the path.

\documentclass{article}
%\url{https://tex.stackexchange.com/q/446941/86}
\usepackage{tikz}
   \usetikzlibrary{patterns}
\usepackage{venndiagram}
%
\begin{document}
\begin{venndiagram2sets}[shade={},tikzoptions={red}]
\begin{scope}[every path/.append style={pattern=north east lines}]
\fillA
\end{scope}
\end{venndiagram2sets}

\end{document}

The shade={} is important. It enables filling (since shade=<colour> becomes fill=<colour> in the code) but doesn't specify a colour. The point here is that the every path style is examined first so putting shade=<actual colour> would override the pattern, while shade=none would disable it. Only shade={} enables the pattern without overwriting it.

---

Update 2018-08-22

Dealing with intersection regions turns out to be a bit tricky due to how clipping works. The following is definitely in the region of a bit of a hack, but it does seem to work. It appends some code to the clip command which removes the options and turns off any other options. I think this deals with everything that can go bad with passing options to a clipping path.

I've also add a helper command that wraps a venn diagram command in a scope so that this technique works.

\documentclass{article}
%\url{https://tex.stackexchange.com/q/446941/86}
\usepackage{tikz}
   \usetikzlibrary{patterns}
\usepackage{venndiagram}
%

\newcommand*\wrapscope[1]{%
  \expandafter\newcommand\csname o#1\endcsname[1][]{%
    \begin{scope}[##1]
    \csname #1\endcsname
    \end{scope}
  }%
}%

\wrapscope{fillA}
\wrapscope{fillACapB}

\makeatletter
\tikzset{
  clip/.append code={%
    \let\tikz@options=\pgfutil@empty
    \tikz@addmode\tikz@mode@fillfalse%
    \tikz@addmode\tikz@mode@drawfalse%
    \tikz@addmode\tikz@mode@doublefalse%
    \tikz@addmode\tikz@mode@boundaryfalse%
    \tikz@addmode\tikz@mode@fade@pathfalse%
    \tikz@addmode\tikz@mode@fade@scopefalse%
  }
}
\makeatother

\begin{document}
\begin{venndiagram2sets}[shade={},tikzoptions={red}]
%\ofillA[every path/.append style={pattern=north east lines}]
\ofillACapB[every path/.append style={pattern=north east lines}]
\end{venndiagram2sets}

\end{document}